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Advances in Nonlinear Analysis

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Hölder gradient estimates for a class of singular or degenerate parabolic equations

Cyril Imbert
  • Department of Mathematics and Applications, CNRS & École Normale Supérieure (Paris), 45 rue d’Ulm, 75005 Paris, France
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/ Tianling Jin
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  • Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
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/ Luis Silvestre
Published Online: 2017-09-06 | DOI: https://doi.org/10.1515/anona-2016-0197

Abstract

We prove interior Hölder estimates for the spatial gradients of the viscosity solutions to the singular or degenerate parabolic equation

ut=|u|κdiv(|u|p-2u),

where p(1,) and κ(1-p,). This includes the from L to C1,α regularity for parabolic p-Laplacian equations in both divergence form with κ=0, and non-divergence form with κ=2-p.

Keywords: Regularity; degenerate parabolic equations; singular parabolic equations

MSC 2010: 35B65; 35K92; 35K65; 35K67

1 Introduction

Let 1<p< and κ(1-p,). We are interested in the regularity of solutions of

ut=|u|κdiv(|u|p-2u).(1.1)

When κ=0, this is the classical parabolic p-Laplacian equation in divergence form. This is the natural case in the context of gradient flows of Sobolev norms. Hölder estimates for the spatial gradient of their weak solutions (in the sense of distribution) were obtained by DiBenedetto and Friedman in [8] (see also Wiegner [26]).

When κ=2-p, equation (1.1) is a parabolic homogeneous p-Laplacian equation. This is the most relevant case for applications to tug-of-war-like stochastic games with white noise; see Peres and Sheffield [22]. This equation has been studied by Garofalo [10], Banerjee and Garofalo [3, 4, 5], Does [9], Manfredi, Parviainen and Rossi [19, 20], Rossi [23], Juutinen [15], Kawohl, Krömer and Kurtz [16], Liu and Schikorra [18], Rudd [24] as well as the second and third authors of this paper [14]. Hölder estimates for the spatial gradient of their solutions were proved in [14]. The solution of this equation is understood in the viscosity sense. The toolbox of methods that one can apply are completely different to the variational techniques used classically for p-Laplacian problems.

Equation (1.1) can be rewritten as

ut=|u|γ(Δu+(p-2)|u|-2uiujuij),(1.2)

where γ=p+κ-2>-1. In this paper, we prove Hölder estimates for the spatial gradients of viscosity solutions to (1.2) for 1<p< and γ(-1,). Therefore, it provides a unified approach for all those γ and p, including the two special cases γ=0 and γ=p-2 mentioned above.

The viscosity solutions to (1.2) with γ>-1 and p>1 fall into the general framework studied by Ohnuma and Sato in [21], which is an extension of the work of Barles and Georgelin [6] and Ishii and Souganidis [13] on the viscosity solutions of singular/degenerate parabolic equations. We postpone the definition of viscosity solutions of (1.2) to Section 5. For r>0, by Qr we denote Br×(-r2,0], where Brn is the ball of radius r centered at the origin.

Theorem 1.1.

Let u be a viscosity solution of (1.2) in Q1, where 1<p< and γ(-1,). Then there exist two constants α(0,1) and C>0, both of which depend only on n, γ, p and uL(Q1), such that

uCα(Q1/2)C.

Also, the following Hölder regularity in time holds:

sup(x,t),(x,s)Q1/2|u(x,t)-u(x,s)||t-s|(1+α)/(2-αγ)C.

Note that (1+α)/(2-αγ)>12 for every α>0 and γ>-1.

Our proof in this paper follows a similar structure as in [14], with some notable differences that we explain below. We use non-divergence techniques in the context of viscosity solutions. The classical variational methods can only be used for γ=p-2, when the equation is in divergence form. Theorem 1.1 tells us that our techniques are in some sense stronger when dealing with the regularity of scalar p-Laplacian-type equations. The weakness of our methods (at least as of now) is that they are ineffective for systems.

The result in [14] has recently been extended to allow for a bounded right-hand side of the equation by Attouchi and Parviainen in [1]. We have not explored the possibility of adding a right-hand side for arbitrary values of the exponent κ.

The greatest difficulty extending the result in [14] to Theorem 1.1 comes from the lack of uniform ellipticity. When γ=0, equation (1.2) is a parabolic equation in non-divergence form with uniformly elliptic coefficients (depending on the solution u). Because of this, in [14], we use the theory developed by Krylov and Safonov, and other classical results, to get some basic uniform a priori estimates. This fact is no longer true for other values of γ. The first step in our proof is to obtain a Lipschitz modulus of continuity. That step uses the uniform ellipticity very strongly in [14]. In this paper, we take a different approach using the method of Ishii and Lions [12] (see also [11, Theorem 5]). Another step where the uniform ellipticity plays a strong role is in a lemma which transfers an oscillation bound in space, for every fixed time, to a space-time oscillation. In this paper, that is achieved through Lemmas 4.4 and 4.5, which are considerably more difficult than their counterpart in [14]. Other, more minor, difficulties include the fact that the non-homogeneous right-hand side forces us to work with a different scaling (see the definition of Qrρ by the beginning of Section 4).

In order to avoid some of the technical difficulties caused by the non-differentiability of viscosity solutions, we first consider the regularized problem (1.3) below, and then obtain uniform estimates so that we can pass to the limit in the end. For ε(0,1), let u be smooth and satisfy

tu=(|u|2+ε2)γ2(δij+(p-2)uiuj|u|2+ε2)uij.(1.3)

We are going to establish Lipschitz estimates and Hölder gradient estimates for u, which will be independent of ε(0,1), in Sections 2, 3 and 4. Then, in Section 5, we recall the definition of viscosity solutions to (1.2) as well as their several useful properties, and prove Theorem 1.1 via approximation arguments. This idea of approximating the problem with a smoother one and proving uniform estimates is very standard.

2 Lipschitz estimates in the spatial variables

The proof of Lipschitz estimates in [14] for γ=0 is based on a calculation that |u|p is a subsolution of a uniformly parabolic equation. We are not able to find a similar quantity for other nonzero γ. The proof we give here is completely different. It makes use of the Ishii–Lions’ method [12]. However, we need to apply this method twice: first we obtain log-Lipschitz estimates, and then use this log-Lipschitz estimates and the Ishii–Lions’ method again to prove Lipschitz estimates. Moreover, the Lipschitz estimates holds for γ>-2 instead of γ>-1.

Lemma 2.1 (Log-Lipschitz estimate).

Let u be a smooth solution of (1.3) in Q4 with γ>-2 and ε(0,1). Then there exist two positive constants L1 and L2 depending only on n, p, γ and uL(Q4) such that for every (t0,x0)Q1 we have

u(t,x)-u(t,y)L1|x-y||log|x-y||+L22|x-x0|2+L22|y-x0|2+L22(t-t0)2

for all t[t0-1,t0] and x,yB1(x0).

Proof.

Without loss of generality, we assume x0=0 and t0=0. It is sufficient to prove that

M:=max-1t0,x,yB1¯{u(t,x)-u(t,y)-L1ϕ(|x-y|)-L22|x|2-L22|y|2-L22t2}

is non-positive, where

ϕ(r)={-rlogrfor r[0,e-1],e-1for re-1.

We assume this is not true and we will exhibit a contradiction. In the rest of the proof, t[-1,0] and x,yB¯1 denote the points realizing the maximum defining M.

Since M0, we have

L1ϕ(|x-y|)+L22(|x|2+|y|2+t2)2uL(Q4).

In particular,

ϕ(δ)2uL(Q4)L1,where δ=|a| and a=x-y,(2.1)

and

|t|+|x|+|y|6uL(Q4)L2.(2.2)

Hence, for L2 large enough depending only on uL(Q4), we can ensure that t(-1,0] and x,yB1. We choose L2 here and fix it for the rest of the proof. Thus, from now on L2 is a constant depending only on uL.

Choosing L1 large, we can ensure that δ (<e-2) is small enough to satisfy

ϕ(δ)2δ.

In this case, (2.1) implies

δuL(Q4)L1.

Since t[-1,0] and x,yB1 realize the supremum defining M, we have that

u(t,x)=L1ϕ(δ)a^+L2x,u(t,y)=L1ϕ(δ)a^-L2y,ut(t,x)-ut(t,y)=L2t,[2u(t,x)00-2u(t,y)]L1[Z-Z-ZZ]+L2I,(2.3)

where

Z=ϕ′′(δ)a^a^+ϕ(δ)δ(I-a^a^)anda^=a|a|=x-y|x-y|.

For zn, we let

A(z)=I+(p-2)zizj|z|2+ε2,

q=L1ϕ(δ)a^, X=2u(t,x) and Y=2u(t,y). By evaluating the equation at (t,x) and (t,y), we have

L2t(|q+L2x|2+ε2)γ2Tr(A(q+L2x)X)-(|q-L2y|2+ε2)γ2Tr(A(q-L2y)Y).(2.4)

Whenever we write C in this proof, we denote a positive constant, large enough depending only on n, p, γ and uL(Q4), which may vary from line to line. Recall that we have already chosen L2 above depending on uL only.

Note that |q|=L1|ϕ(δ)|. By choosing L1 large enough, δ will be small, |ϕ(δ)| will thus be large, and |q|L2. In particular,

|q|2|q+L2x|2|q|and|q|2|q-L2y|2|q|.(2.5)

From (2.3) and the fact that ϕ′′(δ)<0, we have

X=2u(t,x)L1ϕ(δ)δ(I-a^a^)+L2I,-Y=-2u(t,y)L1ϕ(δ)δ(I-a^a^)+L2I.(2.6)

Making use of (2.4), (2.5) and (2.6), we have

Tr(A(q+L2x)X)=(|q+L2x|2+ε2)-γ2L2t+(|q-L2y|2+ε2|q+L2x|2+ε2)γ2Tr(A(q-L2y)Y)-C(|q|-γ+L1ϕ(δ)δ+1).

Therefore, it follows from (2.6) and the ellipticity of A that

|X|C(|q|-γ+L1ϕ(δ)δ+1).(2.7)

Similarly,

|Y|C(|q|-γ+L1ϕ(δ)δ+1).

Let

B(z)=(|z|2+ε)γA(z).

We get from (2.4) and (2.2) the following inequality:

-CTr[B(q+L2x)X]-Tr[B(q-L2y)Y]T1+T2,(2.8)

where

T1=Tr[B(q-L2y)(X-Y)] and T2=|X||B(q+L2x)-B(q-L2y)|.

We first estimate T2. Using successively (2.2), (2.5), (2.7) and the mean value theorem, we get

T2C|X||q|γ-1|x+y|C|X||q|γ-1C(|q|-γ+L1ϕ(δ)δ+1)|q|γ-1C(|q|-1+|q|γδ+|q|γ-1).(2.9)

We now turn to T1. On the one hand, evaluating (2.3) with respect to a vector of the form (ξ,ξ), for all ξd we have

(X-Y)ξξ2L2|ξ|2.(2.10)

On the other hand, when we evaluate (2.3) with respect to (a^,a^), we get

(X-Y)a^a^4L1ϕ′′(δ)+2L2.(2.11)

Inequality (2.10) tells us that all eigenvalues of (X-Y) are bounded above by a constant C. Inequality (2.11) tells us that there is at least one eigenvalue that is less than the negative number 4L1ϕ′′(δ)+2L2. Because of the uniform ellipticity of A, we obtain

T1C|q|γ(L1ϕ′′(δ)+1).

In view of the estimates for T1 and T2, we finally get from (2.8) that

-L1ϕ′′(δ)|q|γC(|q|γ+|q|-1+|q|γδ+|q|γ-1+1),

or equivalently

-L1ϕ′′(δ)C(1+|q|-1-γ+1δ+|q|-1+|q|-γ).(2.12)

Our purpose is to choose L1 large in order to get a contradiction in (2.12).

Recall that we have the estimate δC/L1. From our choice of ϕ, we obtain ϕ(δ)>1 for δ small and -ϕ′′(δ)=1δcL1.

For L1 sufficiently large, since γ>-2,

C(1+|q|-1-γ+|q|-1+|q|-γ)C(1+L1-1-γ+L1-1+L1-γ)c2L12-12L1ϕ′′(δ).

The remaining term is handled because of the special form of the function ϕ. We have

-L1ϕ′′(δ)=L1δ>2Cδ

for L1 sufficiently large.

Therefore, we reached a contradiction. The proof of this lemma is thereby completed. ∎

By letting t=t0 and y=x0 in Lemma 2.1 and since (x0,t0) is arbitrary, we have the following corollary.

Corollary 2.2.

Let u be a smooth solution of (1.3) in Q4 with γ>-2 and ε(0,1). Then there exists a positive constant C depending only on n, γ, p and uL(Q4) such that for every (t,x),(t,y)Q3 and |x-y|<12, we have

|u(t,x)-u(t,y)|C|x-y||log|x-y||.

We shall make use of the above log-Lipschitz estimate and the Ishii–Lions method [12] again to prove the following Lipschitz estimate.

Lemma 2.3 (Lipschitz estimate).

Let u be a smooth solution of (1.3) in Q4 with γ>-2 and ε(0,1). Then there exist two positive constants L1 and L2 depending only on n, p, γ and uL(Q4) such that for every (t0,x0)Q1 we have

u(t,x)-u(t,y)L1|x-y|+L22|x-x0|2+L22|y-x0|2+L22(t-t0)2

for all t[t0-1,t0] and x,yB1/4(x0).

Proof.

The proof of this lemma follows the same computations as that of Lemma 2.1, but we make use of the conclusion of Corollary 2.2 in order to improve our estimate.

Without loss of generality, we assume x0=0 and t0=0. As before, we define

M:=max-1t0,x,yB1{u(t,x)-u(t,y)-L1ϕ(|x-y|)-L22|x|2-L22|y|2-L22t2}

and prove that it is non-positive, where

ϕ(r)={r-12-γ0r2-γ0for r[0,1],1-12-γ0for r1

for some γ0(12,1).

We assume this is not true in order to obtain a contradiction. In the remaining of the proof of the lemma, t[-1,0] and x,yB¯1/4 denote the points realizing the maximum defining M.

For the same reasons as in the proof of Lemma 2.1, inequalities (2.1) and (2.2) also apply in this case. Thus, we can use the same choice of L2 depending on uL only that ensures t(-1,0] and x,yB1.

From Corollary 2.2, we already know that u(t,x)-u(t,y)C|x-y||log|x-y||. Since M0,

L1ϕ(|x-y|)+L22(|x|2+|y|2+t2)C|x-y||log|x-y||.

In particular, we obtain an improvement of (2.2):

|t|+|x|+|y|Cδ|logδ|L2.

This gives us an upper bound for |x+y| that we can use to improve (2):

T2C|X||q|γ-1|x+y|C(|q|-1+|q|γδ+|q|γ-1)δ|logδ|.

The estimate for T1 stays unchanged. Hence, (2.12) becomes

-L1ϕ′′(δ)C(1+δ|logδ|(|q|-1+|q|-1-γ+1δ+|q|-γ)).

Recall that |q|=L1ϕ(δ)L1/2 and ϕ′′(δ)=(γ0-1)δ-γ0. Then

L1δ-γ0C(1+δ|logδ|(1+L1-1+L1-1-γ+δ-1+L1-γ)).

The term +1 inside the innermost parenthesis is there just to ensure that the inequality holds both for γ<0 and γ>0. Recalling that δ<C/L1, we obtain an inequality in terms of L1 only:

L11+γ0C(1+L1-12logL1(1+L1-1+L1-1-γ+L1+L1-γ))

Choosing L1 large, we arrive at a contradiction given that 1+γ0>max(12,-12-γ) since γ0>12 and γ>-2. ∎

Again, by letting t=t0 and y=x0 in Lemma 2.3 and since (x0,t0) is arbitrary, we have the following corollary.

Corollary 2.4.

Let u be a smooth solution of (1.3) in Q4 with γ>-2 and ε(0,1). Then there exists a positive constant C depending only on n, γ, p and uL(Q4) such that for every (t,x),(t,y)Q3 and |x-y|<1,

|u(t,x)-u(t,y)|C|x-y|.

3 Hölder estimates in the time variable

Using the Lipschitz continuity in x and a simple comparison argument, we show that the solution of (1.3) is Hölder continuous in t.

Lemma 3.1.

Let u be a smooth solution of (1.3) in Q4 with γ>-1 and ε(0,1). Then there holds

supts,(t,x),(s,x)Q1|u(t,x)-u(s,x)||t-s|1/2C,

where C is a positive constant depending only on n, p, γ and uL(Q4).

Remark 3.2.

Deriving estimates in the time variable for estimates in the space variable by maximum principle techniques is classical. As far as viscosity solutions are concerned, the reader is referred to [2, Lemma 9.1, p. 317] for instance.

Proof.

Let β=max(2,(2+γ)/(1+γ)). We claim that for all t0[-1,0) and η>0 there exist L1>0 and L2>0 such that

u(t,x)-u(t0,0)η+L1(t-t0)+L2|x|β=:φ(t,x)for all (t,x)[t0,0]×B¯1.(3.1)

We first choose L22uL(Q3) such that (3.1) holds true for xB1. We will next choose L2 such that (3.1) holds true for t=t0. In this step, we shall use Corollary 2.4 to find that u is Lipschitz continuous with respect to the spatial variables. From Corollary 2.4, uL(Q3) is bounded depending on uL(Q4) only. It is enough to choose

uL(Q3)|x|η+L2|x|β,

which holds true if

L2uL(Q3)βηβ-1.

We finally choose L1 such that the function φ(t,x) is a supersolution of an equation which u is a solution of. Inequality (3.1) thus follows from the comparison principle. We use a slightly different equation depending on whether γ0 or γ>0.

Let us start with the case γ0. In this case, we will prove that φ is a supersolution of the nonlinear equation (1.3). That is,

φt-(ε2+|φ|2)γ2(δij+(p-2)φiφjε2+|φ|2)φij>0.(3.2)

In order to ensure this inequality, we choose L1 so that

L1>(p-1)|φ|γ|D2φ|(ε2+|φ|2)γ2(δij+(p-2)φiφjε2+|φ|2)φij.

We chose the exponent β so that when γ0, |φ|γ|D2φ|=CL11+γ for some constant C depending on n and γ. Thus, we must choose L1=CL21+γ in order to ensure (3.2).

Therefore, still for the case γ0, β=(2+γ)/(1+γ) and for any choice of η>0, using the comparison principle, we have

u(t,0)-u(t0,0)η+C(η(1-β)uL(Q3)β+2uL(Q3)+ε)γ+1(t-t0)η+Cη-1uL(Q3)γ+2|t-t0|+C(uL(Q3)+ε)γ+1|t-t0|.

By choosing η=uL(Q3)γ/2+1|t-t0|1/2, it follows that for t(t0,0],

u(t,0)-u(t0,0)C(uL(Q3))γ+22|t-t0|12+C(uL(Q3)+ε)γ+1|t-t0|.

The lemma is then concluded in the case γ0.

Let us now analyze the case γ>0. In this case, we prove that φ is a supersolution to a linear parabolic equation whose coefficients depend on u. That is,

φt-(ε2+|u|2)γ2(δij+(p-2)uiujε2+|u|2)φij>0.

Since γ>0 and u is known to be bounded after Corollary 2.4, we can rewrite the equation assumption as

φt-aij(t,x)φij>0,(3.3)

where the coefficients aij(t,x) are bounded by

|aij(t,x)|C(ε+uL(Q3))γ.

Since γ>0, we pick β=2 and D2φ is a constant multiple of L2. In particular, we ensure that (3.3) holds if

L1>C(ε+uL(Q3))γL2.

Therefore, for the case γ>0, β=2 and for any choice of η>0, by using the comparison principle,

u(t,0)-u(t0,0)η+C(ε+uL(Q3))γ(η-1uL(Q3)2+uL(Q3))(t-t0).

Choosing

η=(ε+uL(Q3))γ2+1(t-t0)12,

we obtain,

u(t,0)-u(t0,0)C(ε+uL(Q3))γ2+1(t-t0)12+C(ε+uL(Q3))γuL(Q3)(t-t0).

This finishes the proof for γ>0 as well. ∎

4 Hölder estimates for the spatial gradients

In this section, we assume that γ>-1, so that Corollary 2.4 and Lemma 3.1 hold, that is, the solution of (1.3) in Q2 has uniform interior Lipschitz estimates in x and uniform interior Hölder estimates in t, both of which are independent of ε(0,1). For ρ,r>0, we denote

Qr=Br×(-r2,0],Qrρ=Br×(-ρ-γr2,0].

This same family of cylinders Qrρ was used in [8]. They are the natural ones that correspond to the two-parameter family of scaling of the equation. Indeed, if u solves (1.3) in Qrρ and we let v(x,t)=1rρu(rx,r2ρ-γt), then

vt(t,x)=(|v|2+ε2ρ-2)γ2(Δv+(p-2)vivj|v|2+ε2ρ-2vij)in Q1.

If we choose ρuL(Q1)+1, we may assume that the solution of (1.3) satisfies |u|1 in Q1.

We are going to show that u is Hölder continuous in space-time at the point (0,0). The idea of the proof in this step is similar to that in [14]. First we show that if the projection of u onto the direction e𝕊n-1 is away from 1 in a positive portion of Q1, then ue has improved oscillation in a smaller cylinder.

Lemma 4.1.

Let u be a smooth solution of (1.3) with ε(0,1) such that |u|1 in Q1. For every 12<<1 and μ>0, there exists τ1(0,14) depending only on μ,n and there exist τ,δ>0 depending only on n, p, γ, μ and such that for arbitrary eSn-1 if

|{(x,t)Q1:ue}|>μ|Q1|,

then

ue<1-δin Qτ1-δ

and Qτ1-δQτ1.

Proof.

Let

aij(q)=(|q|2+ε2)γ2(δij+(p-2)qiqj|q|2+ε2),qn,(4.1)

and denote

aij,m=aijqm.

Differentiating (1.3) in xk, we have

(uk)t=aij(uk)ij+aij,muij(uk)m.

Then

(ue-)t=aij(ue-)ij+aij,muij(ue-)m,

and for

v=|u|2

we have

vt=aijvij+aij,muijvm-2aijukiukj.

For ρ=4, let

w=(ue-+ρ|u|2)+.

Then in the region Ω+={(x,t)Q1:w>0} we have

wt=aijwij+aij,muijwm-2ρaijukiukj.

Since |u|>2 in Ω+, we have

|aij,m|{c(p,n,γ)-1if γ0,c(p,n,γ)γ-1if γ<0,

in Ω+, where c(p,n,γ) is a positive constant depending only on p, n and γ. By the Cauchy–Schwarz inequality, it follows that

wtaijwij+c1()|w|2in Ω+,

where

c1()={c0-γ-3if γ0,c02γ-3if γ<0

for some constant c0>0 depending only on p, γ and n. Therefore, it satisfies in the viscosity sense that

wta~ijwij+c1()|w|2in Q1,

where

a~ij(x)={aij(u(x))if xΩ+,δijelsewhere.

Notice that since (12,1), the coefficient a~ij is uniformly elliptic with ellipticity constants depending only on p and γ. We can choose c2()>0 depending only on p, γ, n and such that if we let

W=1-+ρ

and

w¯=1c2(1-ec2(w-W)),

then we have

w¯ta~ijw¯ijin Q1

in the viscosity sense. Since WsupQ1w, we obtain w¯0 in Q1.

If ue, then w¯(1-ec2(-1))/c2. Therefore, it follows from the assumption that

|{(x,t)Q1:w¯(1-ec2(-1))c2}|>μ|Q1|.

By [14, Proposition 2.3], there exist τ1>0 depending only on μ and n, and ν>0 depending only on μ, , n, γ and p such that

w¯νin Qτ1.

Meanwhile, we have

w¯W-w.

This implies that

W-wνin Qτ1.

Therefore, we have

ue+ρ|u|21+ρ-νin Qτ1.

Since |ue||u|, we have

ue+ρ(ue)21+ρ-νin Qτ1.

Therefore, remarking that ν1+ρ, we have

ue-1+1+4ρ(1+ρ-ν)2ρ1-δin Qτ1

for some δ>0 depending only on p, γ, μ, and n. Finally, we can choose τ=τ1 if γ<0 and τ=τ1(1-δ)γ/2 if γ0 such that Qτ1-δQτ1. ∎

Note that our choice of τ and δ above implies that

τ<(1-δ)γ2when γ0.

In the rest of the paper, we will choose τ even smaller such that

τ<(1-δ)1+γfor all γ>-1.(4.2)

This fact will be used in the proof of Theorem 4.8.

In case we can assume that Lemma 4.1 holds in all directions eB1, then it effectively implies a reduction in the oscillation of u in a smaller parabolic cylinder. If such an improvement of oscillation takes place at all scales, it leads to the Hölder continuity of u at (0,0) by iteration and scaling. The following corollary describes this favorable case in which the assumption of the previous lemma holds in all directions.

Corollary 4.2.

Let u be a smooth solution of (1.3) with ε(0,1) such that |u|1 in Q1. For every 0<<1 and μ>0, there exist τ(0,14) depending only on μ and n, and δ>0 depending only on n, p, γ, μ and , such that for every nonnegative integer klogε/log(1-δ) if

|{(x,t)Qτi(1-δ)i:ue(1-δ)i}|>μ|Qτi(1-δ)i|for all e𝕊n-1 and i=0,,k,(4.3)

then

|u|<(1-δ)i+1in Qτi+1(1-δ)i+1 for all i=0,,k.

Remark 4.3.

Note that we can further impose on δ that δ<12 and δ<1-τ.

Proof.

When i=0, it follows from Lemma 4.1 that ue<1-δ in Qτ for all e𝕊n-1. This implies that |u|<1-δ in Qτ1-δ.

Suppose this corollary holds for i=0,,k-1. We are going prove it for i=k. Let

v(x,t)=1τk(1-δ)ku(τkx,τ2k(1-δ)-kγt).

Then v satisfies

vt=(|v|2+ε2(1-δ)2k)γ2(Δv+(p-2)vivj|v|2+ε2(1-δ)-2kvij)in Q1.

By the induction hypothesis, we also know that |v|1 in Q1, and

|{(x,t)Q1:ve}|>μ|Q1|for all e𝕊n-1.

Notice that ε(1-δ)k. Therefore, by Lemma 4.1 we have

ve1-δin Qτ1-δ for all e𝕊n-1.

Hence, |v|1-δ in Qτ1-δ. Consequently,

|u|<(1-δ)k+1in Qτk+1(1-δ)k+1.

Unless u(0,0)=0, the above iteration will inevitably stop at some step. There will be a first value of k where the assumptions of Corollary 4.2 do not hold in some direction e𝕊n-1. This means that u is close to some fixed vector in a large portion of Qτk(1-δ)k. We then prove that u is close to some linear function, from which the Hölder continuity of u will follow by applying a result from [25].

Having u close to a vector e for most points tells us that for every fixed time t the function u(x,t) will be approximately linear. However, it does not say anything about how u varies with respect to time. We must use the equation in order to prove that the function u(x,t) will be close to some linear function uniformly in t. This is the main purpose of the following set of lemmas.

Lemma 4.4.

Let uC(Q¯1) be a smooth solution of (1.3) with γ>-1 and ε(0,1) such that |u|M in Q1. Let A be a positive constant. Assume that for all t[-1,0] we have

oscB1u(,t)A.

Then

oscQ1u{CAif γ0,C(A+A1+γ)if -1<γ<0,

where C is a positive constant depending only on M, γ, p and the dimension n.

Proof.

When γ0, for the aij in (4.1) we have |aij|Λ:=(M2+1)γ/2max(p-1,1), and therefore the conclusion follows from the same proof of [14, Lemma 4.3].

When γ(-1,0), we choose different comparison functions from [14]. Let

w¯(x,t)=a¯+ΛA1+γt+2A|x|β,w¯(x,t)=a¯-ΛA1+γt-2A|x|β,

where β=(2+γ)/(1+γ) and Λ is to be fixed later. As far as a¯ and a¯ are concerned, a¯ is chosen so that w¯(,-1)u(,-1) in B1 and w¯(x¯,-1)=u(x¯,-1) for some x¯B¯1, and a¯ is chosen so that w¯(,-1)u(,-1) in B1 and w¯(x¯,-1)=u(x¯,-1) for some x¯B¯1. This implies that

a¯-a¯=u(x¯,-1)-u(x¯,-1)+2ΛA1+γ-2A|x¯|2-2A|x¯|2A+2ΛA1+γ.

Notice that β>2 since γ(-1,0). We now remark that if Λ is chosen as Λ=(2β)γ+1(β-1)pn2+1, then the first inequality

ΛA1+γ((2Aβ|x|β-1)2+ε2)γ2pn22Aβ(β-1)|x|β-2(2β)γ+1(β-1)pn2A1+γ

(we used that γ<0) cannot hold true for xB1. This implies that w¯ is a strict supersolution of the equation satisfied by u. Similarly, w¯ is a strict subsolution.

We claim that

w¯uin Q1  and  w¯uin Q1.

We only justify the first inequality since we can proceed similarly to get the second one. Suppose that the first inequality is false. Let m=-infQ1(w¯-u)>0 and (x0,t0)Q¯1 be such that m=u(x0,t0)-w¯(x0,t0). Then w¯+mu in Q1 and w¯(x0,t0)+m=u(x0,t0). By the choice of a¯, we know that t0>-1. If x0B1, then

2A=(w¯(x0,t0)+m)-(w¯(0,t0)+m)u(x0,t0)-u(0,t0)oscB1u(,t0)A,

which is impossible. Therefore, x0B1. But this is not possible since w¯ is a strict supersolution of the equation satisfied by u. This proves the claim.

Therefore, we have

oscQ1usupQ1w¯-infQ1w¯a¯-a¯+4A=2ΛAγ+1+5A.

Lemma 4.5.

Let uC(Q¯1) be a smooth solution of (1.3) with γR and ε(0,1). Let eSn-1 and 0<δ<18. Assume that for all t[-1,0] we have

oscxB1(u(x,t)-xe)δ.

Then

osc(x,t)Q1(u(x,t)-xe)Cδ,

where C is a positive constant depending only on γ, p and the dimension n.

Proof.

Let

w¯(x,t)=a¯+xe+Λδt+2δ|x|2,w¯(x,t)=a¯+xe-Λδt-2δ|x|2,

where Λ>0 will be fixed later, a¯ is chosen so that w¯(,-1)u(,-1) in B1 and w¯(x¯,-1)=u(x¯,-1) for some x¯B¯1, and a¯ is chosen so that w¯(,-1)u(,-1) in B1 and w¯(x¯,-1)=u(x¯,-1) for some x¯B¯1. This implies that

a¯-a¯=u(x¯,-1)-x¯e-(u(x¯,-1)-x¯e)+2Λδ-2δ|x¯|2-2δ|x¯|2(2Λ+1)δ.

For every xB¯1 and t[-1,0], since δ<18, we have

|w¯(x,t)||e|-4δ|x|12,|w¯(x,t)||e|-4δ|x|12.

Similarly, |w¯(x,t)|32 and |w¯(x,t)|32. Therefore, using the notation from (4.1), there is a constant A0 (depending on p and γ) so that

aij(w¯(x,t))A0I  and  aij(w¯(x,t))A0I.

We choose Λ=5nA0. We claim that

w¯uin Q1  and  w¯uin Q1.

We only justify the first inequality since we can proceed similarly to get the second one. Suppose that the first inequality is false. Let m=-infQ1(w¯-u)>0 and (x0,t0)Q¯1 be such that m=u(x0,t0)-w¯(x0,t0). Then w¯+mu in Q1 and w¯(x0,t0)+m=u(x0,t0). By the choice of a¯, we know that t0>-1. If x0B1, then

2δ=(w¯(x0,t0)+m)-x0e-(w¯(0,t0)+m)u(x0,t0)-x0e-u(0,t0)oscxB1(u(x,t0)-xe)δ,

which is impossible. Hence, x0B1. Therefore, we have the classical relations

u(x0,t0)=w¯(x0,t0)+m,u(x0,t0)=w¯(x0,t0)B¯3/2B1/2,D2u(x0,t0)D2w¯(x0,t0)=4δI,tu(x0,t0)tw¯(x0,t0)=Λδ.

It follows that

ut(x0,t0)-aij(u(x0,t0))iju(x0,t0)w¯t(x0,t0)-aij(w¯(x0,t0))ijw¯(x0,t0)>0,

which is a contradiction. This proves the claim.

Therefore, we have

osc(x,t)Q1(u(x,t)-xe)supQ1(w¯-xe)-infQ1(w¯-xe)a¯-a¯+4δ=(2Λ+5)A.

Lemma 4.6.

Let η be a positive constant and let u be a smooth solution of (1.3) with γ>-1 and ε(0,1) such that |u|1 in Q1. Assume

|{(x,t)Q1:|u-e|>ε0}|ε1

for some eSn-1 and two positive constants ε0, ε1. Then if ε0 and ε1 are sufficiently small, there exists a constant aR such that

|u(x,t)-a-ex|η for all (x,t)Q1/2.

Here, both ε0 and ε2 depend only on n, p, γ and η.

Proof.

Let

f(t):=|{xB1:|u(x,t)-e|>ε0}|.

By the assumptions and Fubini’s theorem, we have that -10f(t)𝑑tε1. For E:={t(-1,0):f(t)ε1}, we obtain

|E|1ε1Ef(t)𝑑t1ε1-10f(t)𝑑tε1.

Therefore, for all t(-1,0]E with |E|ε1 we have

|{xB1:|u(x,t)-e|>ε0}|ε1.(4.4)

It follows from (4.4) and Morrey’s inequality that for all t(-1,0]E we have

oscB1/2(u(,t)-ex)C(n)u-eL2n(B1)C(n)(ε0+ε114n),(4.5)

where C(n)>0 depends only on n.

Meanwhile, since |u|1 in Q1, we have that oscB1u(,t)2 for all t(-1,0]. Therefore, applying Lemma 4.4, we have that oscQ1uC for some constant C. Note that u(t,x)-u(0,0) also satisfies (1.3) and

u(t,x)-u(0,0)L(Q1)oscQ1uC.

By applying Lemma 3.1 to u(t,x)-u(0,0), we have

supts,(t,x),(s,x)Q1|u(t,x)-u(s,x)||t-s|1/2C.

Therefore, by (4.5) and the fact that |E|ε1 we obtain

oscB1/2(u(,t)-ex)C(ε0+ε114n+ε114)

for all t(-14,0] (that is, including tE). If ε0 and ε1 are sufficiently small, we obtain from Lemma 4.5 that

oscQ1/2(u-ex)C(ε0+ε114n+ε114).

Hence, if ε0 and ε1 are sufficiently small, there exists a constant a such that

|u(t,x)-a-ex|η for all (x,t)Q1/2.

Theorem 4.7 (Regularity of small perturbation solutions).

Let u be a smooth solution of (1.3) in Q1. For each β(0,1), there exist two positive constants η (small) and C (large), both of which depend only on β, n, γ and p, such that if |u(x,t)-L(x)|η in Q1 for some linear function L of x satisfying 12|L|2, then

u-LC2,β(Q1/2)C.

Proof.

Since L is a solution of (1.3), the conclusion follows from [25, Corollary 1.2]. ∎

Now we are ready to prove the following Hölder gradient estimate.

Theorem 4.8.

Let u be a smooth solution of (1.3) with ε(0,1) and γ>-1 such that |u|1 in Q1. Then there exist two positive constants α and C depending only on n, γ and p such that

|u(x,t)-u(y,s)|C(|x-y|α+|t-s|α2-αγ)

for all (x,t),(y,s)Q1/2. Also, there holds

|u(x,t)-u(x,s)|C|t-s|1+α2-αγ

for all (x,t),(x,s)Q1/2.

Proof.

We first show the Hölder estimate of u at (0,0) and the Hölder estimate in t at (0,0).

Let η be the one from Theorem 4.7 with β=12, and for this η let ε0, ε1 be two sufficiently small positive constants so that the conclusion of Lemma 4.6 holds. For =1-ε02/2 and μ=ε1/|Q1| if

|{(x,t)Q1:ue}|μ|Q1|for any e𝕊n-1,

then

|{(x,t)Q1:|u-e|>ε0}|ε1.

This is because if |u(x,t)-e|>ε0 for some (x,t)Q1, then

|u|2-2ue+1ε02.

Since |u|1, we have

ue1-ε022.

Therefore, if =1-ε02/2 and μ=ε1/|Q1|, then

{(x,t)Q1:|u-e|>ε0}{(x,t)Q1:ue},(4.6)

from which it follows that

|{(x,t)Q1:|u-e|>ε0}||{(x,t)Q1:ue}|μ|Q1|ε1.

Let τ and δ be the constants from Corollary 4.2. We denote by [logε/log(1-δ)] the integer part of logε/log(1-δ). Let k be either [logε/log(1-δ)] or the minimum nonnegative integer such that condition (4.3) does not hold, whichever is smaller. Then it follows from Corollary 4.2 that for all =0,1,,k we have

|u(x,t)|(1-δ)in Qτ(1-δ).

Then for

(x,t)Qτ(1-δ)Qτ+1(1-δ)+1

we obtain

|u(x,t)|(1-δ)C(|x|α+|t|α2-αγ),

where

C=11-δandα=log(1-δ)logτ.

Thus,

|u(x,t)-q|C(|x|α+|t|α2-αγ)in Q1Qτk+1(1-δ)k+1(4.7)

for every qn such that |q|(1-δ)k. Note that when γ0, it follows from (4.2) that

2-αγ>0andα2-αγ<12.

For =0,1,,k, let

u(x,t)=1τ(1-δ)u(τx,τ2(1-δ)-γt).(4.8)

Then |u(x,t)|1 in Q1, and

tu=(|u|2+ε2(1-δ)-2)γ2(δij+(p-2)iuju|u|2+ε2(1-δ)-2)ijuin Q1.(4.9)

Notice that ε2(1-δ)-2ε2(1-δ)-2k1. By Lemma 4.4, we have

oscQ1uC,

and thus

oscQτ(1-δ)uCτ(1-δ).(4.10)

Let v=uk.

Case 1: k=[logε/log(1-δ)]. Then we have (1-δ)k+1<ε(1-δ)k, and thus 12<1-δ<ε(1-δ)-k1. Therefore, when =k, equation (4.9) is a uniformly parabolic quasilinear equation with smooth and bounded coefficients. By the standard quasilinear parabolic equation theory (see, e.g., [17, Theorem 4.4, p. 560]) and Schauder estimates, there exists bn, |b|1, such that

|v(x,t)-b|C(|x|+|t|12)C(|x|α+|t|α2-αγ)in Qτ1-δQ1/4

and

|tv|Cin Qτ1-δQ1/4,

where C>0 depends only on γ, p and n, and we used that α2-αγ12. Rescaling back, we have

|u(x,t)-(1-δ)kb|C(|x|α+|t|α2-αγ)in Qτk+1(1-δ)k+1(4.11)

and

|u(x,t)-u(x,0)|Cτ-k(1-δ)k(γ+1)|t|in Qτk+1(1-δ)k+1.(4.12)

Then we can conclude from (4.7) and (4.11) that

|u(x,t)-q|C(|x|α+|t|α2-αγ)in Q1/2,

where C>0 depends only on γ, p and n. From (4.12) we obtain that for |t|τ2m(1-δ)-mγ with mk+1,

|u(0,t)-u(0,0)|Cτ-k(1-δ)k(γ+1)τ2m(1-δ)-mγCτm(1-δ)m,(4.13)

where in the last inequality we have used (4.2). From (4.10) and (4.13) we have

|u(0,t)-u(0,0)|C|t|β

for all t(-14,0], where β is chosen such that

τ(1-δ)=(τ2(1-δ)-γ)β.

That is,

β=1+α2-αγ.(4.14)

Note that β>12 if γ>-2.

Case 2: k<[logε/log(1-δ)]. Then

|{(x,t)Qτk(1-δ)k:ue(1-δ)k}|μ|Qτk(1-δ)k|for some e𝕊n-1.

Also,

|u|<(1-δ)in Qτ(1-δ) for all =0,1,,k.

Recall v=uk as defined in (4.8), which satisfies (4.9) with =k. Then |v|1 in Q1, and

|{(x,t)Q1:ve}|μ|Q1|for some e𝕊n-1.

Consequently, using (4.6), we get

|{(x,t)Q1:|v-e|>ε0}|ε1.

It follows from Lemma 4.6 that there exists a such that

|v(x,t)-a-ex|η for all (x,t)Q1/2.

By Theorem 4.7, there exists bn such that

|v-b|C(|x|+|t|)for all (x,t)Qτ1-δQ1/4

and

|tv|Cin Qτ1-δQ1/4.

Rescaling back, we have

|u(x,t)-(1-δ)kb|C(|x|α+|t|α2-αγ)in Qτk+1(1-δ)k+1

and

|u(x,t)-u(x,0)|Cτ-k(1-δ)k(γ+1)|t|in Qτk+1(1-δ)k+1.

Together with (4.7) and (4.10), we can conclude as in case 1 that

|u(x,t)-q|C(|x|α+|t|α2-αγ)in Q1/2

and

|u(0,t)-u(0,0)|C|t|β

for all t(-14,0], where C>0 depends only on γ, p and n.

In conclusion, we have proved that there exist qn with |q|1, and two positive constants α and C depending only on γ, p and n such that

|u(x,t)-q|C(|x|α+|t|α2-αγ)for all (x,t)Q1/2

and

|u(0,t)-u(0,0)|C|t|βfor t(-14,0],

where β is given in (4.14). Then the conclusion follows from standard translation arguments. ∎

5 Approximation

As mentioned in the introduction, the viscosity solutions to

ut=|u|γ(Δu+(p-2)|u|-2uiujuij)in Q1(5.1)

with γ>-1 and p>1 fall into the general framework studied by Ohnuma and Sato in [21], which is an extension of the work of Barles and Georgelin [6] and Ishii and Souganidis [13] on the viscosity solutions of singular/degenerate parabolic equations. Let us recall from [21] the definition of viscosity solutions to (5.1).

We denote

F(u,2u)=|u|γ(Δu+(p-2)|u|-2uiujuij).

Let be the set of functions fC2([0,)) satisfying

f(0)=f(0)=f′′(0)=0,f′′(r)>0for all r>0

and

lim|x|0,x0F(g(x),2g(x))=lim|x|0,x0F(-g(x),-2g(x))=0,where g(x)=f(|x|).

This set is not empty when γ>-1 and p>1 since f(r)=rβ for any β>max((γ+2)/(γ+1),2). Moreover, if f, then λf for all λ>0.

Because equation (5.1) may be singular or degenerate, one needs to choose the test functions properly when defining viscosity solutions. A function φC2(Q1) is admissible, which is denoted as φ𝒜, if for every z^=(x^,t^)Q1 such that φ(z^)=0 there exist δ>0, f and ωC([0,)) satisfying ω0 and limr0ω(r)r=0 such that for all z=(x,t)Q1, |z-z^|<δ, we have

|φ(z)-φ(z^)-φt(z^)(t-t^)|f(|x-x^|)+ω(|t-t^|).

Definition 5.1.

An upper (resp. lower) semi-continuous function u in Q1 is called a viscosity subsolution (resp. supersolution) of (5.1) if for every admissible φC2(Q1) such that u-φ has a local maximum (resp. minimum) at (x0,t0)Q1, the following conditions hold:

φt(resp. )|φ|γ(Δφ+(p-2)|φ|-2φiφjφij)at (x0,t0) when φ(x0,t0)0

and

φt(resp. )0at (x0,t0) when φ(x0,t0)=0.

A function uC(Q1) is called a viscosity solution of (1.1), if it is both a viscosity subsolution and a viscosity supersolution.

We shall use two properties about the viscosity solutions defined above. The first one is the comparison principle for (5.1), which is [21, Theorem 3.1].

Theorem 5.2 (Comparison principle).

Let u and v be a viscosity subsolution and a viscosity supersolution of (5.1) in Q1, respectively. If uv on pQ1, then uv in Q¯1.

The second one is the stability of viscosity solutions of (5.1), which is an application of [21, Theorem 6.1]. Its application to equation (5.1) with γ=0 and 1<p2 is given in [21, Proposition 6.2] with detailed proof. It is elementary to check that it applies to (5.1) for all γ>-1 and all p>1 (which was also pointed out in [21]).

Theorem 5.3 (Stability).

Let {uk} be a sequence of bounded viscosity subsolutions of (1.3) in Q1 with εk0 such that εk0 and uk converges locally uniformly to u in Q1. Then u is a viscosity subsolution of (5.1) in Q1.

Now we shall use the solution of (1.3) to approximate the solution of (5.1). Since p>1, the following lemma ensues from classical quasilinear equations theory (see, e.g., [17, Theorem 4.4, p. 560]) and the Schauder estimates.

Lemma 5.4.

Let gC(pQ1). For ε>0, there exists a unique solution uεC(Q1)C(Q¯1) of (1.3) with p>1 and γR such that uε=g on pQ1.

The last ingredient we need in the proof of Theorem 1.1 is the following continuity estimate up to the boundary for the solutions of (1.3). Its proof is given in Appendix A. For two real numbers a and b, we denote ab=max(a,b) and ab=min(a,b).

Theorem 5.5 (Boundary estimates).

Let uC(Q¯1)C(Q1) be a solution of (1.3) with γ>-1 and ε(0,1). Let φ:=u|pQ1 and let ρ be a modulus of continuity of φ. Then there exists another modulus of continuity ρ* depending only on n, γ, p, ρ and φL(pQ1) such that

|u(x,t)-u(y,s)|ρ*(|x-y||t-s|)

for all (x,t),(y,s)Q¯1.

Proof of Theorem 1.1.

Given Theorem 4.8, Theorem 5.2, Theorem 5.3, Lemma 5.4 and Theorem 5.5, the proof of Theorem 1.1 is identical to that of [14, Theorem 1]. ∎

A Proof of Theorem 5.5

We will adapt some arguments in [7] to prove Theorem 5.5. In the following, c denotes some positive constant depending only on n, γ and p, which may vary from line to line. Denote

Fε(u,2u)=(|u|2+ε2)γ2(δij+(p-2)uiuj|u|2+ε2)uij.

Lemma A.1.

For every zB1, there exists a function WzC(B¯1) such that Wz(z)=0 and Wz>0 in B¯1{z}, and

Fε(Wz,2Wz)-1in B1.

Proof.

Let zB1. Let f(r)=(r-1)+ and wz(x)=f(|x-2z|). Then for xB1 we have

Fε(wz,2wz)=(f2+ε2)γ2((1+(p-2)f2f2+ε2)f′′+n-1|x-2z|f).

Then there exists δ>0 depending only on n, γ and p such that for xB1B1+δ(2z) we have

Fε(wz,2wz)-1.

For

σ=2nmin(p-1,1)+2anda>0,

let

Gz(x)=a(2σ-1|x-2z|σ).

Then Gz(x)a(2σ-1) in B1. Also, for r=|x-2z| and xB1 we have

Fε(Gz,2Gz)=a(σ2r-2σ-2+ε2)γ2((1+(p-2)σ2σ2+ε2r2σ+2)σ(-σ-1)r-σ-2+(n-1)σr-σ-2)-a2σr-σ-2(σ2r-2σ-2+ε2)γ2{-a23-σ-2-γ(σ+1)σ1+γif γ0,-a23-σ-2(σ2+1)γ2σif γ<0,

where in the first inequality we used the choice of σ. Then we choose a such that

a(2σ-1|1+δ|σ)=δ2.

Since wz(z)=0 and Gz(z)>0, the function

Wz(x)={Gz(x)for xB¯1,|x-2z|1+δ,min(Gz(x),wz(x))for xB¯1,|x-2z|1+δ,

agrees with wz in a neighborhood of z (relative to B¯1). Also, because of the choice of a, the function Wz agrees with Gz when xB¯1 and |x-2z|1+δ~ for some δ~(0,δ). Moreover,

Fε(Wz,2Wz)-κ

for some constant κ>0 depending only on n, γ and p. By multiplying a large positive constant to Wz, we finish the proof of this lemma. ∎

Lemma A.2.

For every (z,τ)pQ1, there exists Wz,τC(Q¯1) such that Wz,τ(z,τ)=0, Wz,τ>0 in Q¯1{(z,τ)} and

tWz,τ-Fε(Wz,τ,2Wz,τ)1in Q1.

Proof.

For τ>-1 and xB1,

Wz,τ(x,t)=(t-τ)22+2Wz

is a desired function, where Wz is the one from Lemma A.1. For τ=-1 and xB1, we let

Wz,τ(x,t)=A(t+1)+|x-z|β,

where

β=max(γ+2γ+1,2).

If we choose A>0 large, which depends only on n, γ and p, then Wz,τ will be a desired function. ∎

For two real numbers a and b, we denote ab=max(a,b) and ab=min(a,b).

Theorem A.3.

Let uC(Q¯1)C(Q1) be a solution of (1.3) with γ>-1 and ε(0,1). Let φ:=u|pQ1 and let ρ be a modulus of continuity of φ. Then there exists another modulus of continuity ρ* depending only on n, γ, p and ρ such that

|u(x,t)-u(y,s)|ρ~(|x-y||t-s|)

for all (x,t)Q¯1 and (y,s)pQ1.

Proof.

For every κ>0 and (z,τ)pQ1, let

Wκ,z,τ(x,t)=φ(z,τ)+κ+MκWz,τ(x,t),

where Mκ>0 is chosen so that

φ(z,τ)+κ+MκWz,τ(y,s)φ(y,s)for all (y,s)pQ1.

Indeed,

Mk=inf(y,s)pQ1,(y,s)(z,τ)(ρ(|z-y||τ-s|)-κ)+Wz,τ(y,s)

would suffice, and is independent of the choice of (z,τ). Finally, let

W(x,t)=infκ>0,(z,τ)pQ1Wκ,z,τ(x,t).

Note that for every κ>0 and (z,τ)pQ1,

W(x,t)-φ(z,τ)Wκ,z,τ(x,t)-φ(z,τ)κ+MκWz,τ(x,t)κ+Mκ(Wz,τ(x,t)-Wz,τ(z,τ))κ+Mκω(|z-x||τ-t|),

where ω is the modulus of continuity for Wz,τ, which is evidently independent of (z,τ). Let

ρ~(r)=infκ>0(κ+Mκω(r))

for all r0. Then ρ~ is a modulus of continuity, and

W(x,t)-φ(z,τ)ρ~(|z-x||τ-t|) for all (x,t)Q¯1,(z,τ)pQ1.

By Lemma A.2, Wκ,z,τ is a supersolution of (1.3) for every κ>0 and (z,τ)pQ1, and therefore W is also a supersolution of (1.3). By the comparison principle,

u(x,t)-φ(z,τ)W(x,t)-φ(z,τ)ρ~(|z-x||τ-t|)

for all (x,t)Q¯1 and (z,τ)pQ1.

Similarly, one can show that u(x,t)-φ(z,τ)-ρ~(|z-x||τ-t|) for all (x,t)Q¯1 and (z,τ)pQ1. This finishes the proof of this theorem. ∎

Proof of Theorem 5.5.

By the maximum principle, we have that

M:=uL(Q1)=φL(pQ1).

Let (x,t),(y,s)Q1, and assume that ts. Let x0 be such that |x-x0|=1-|x|=r. Let ρ~ be the one from the conclusion of Theorem A.3. Without loss of generality, we may assume that 2M+2ρ~(r)r for all r[0,2] (e.g., replacing ρ~(r) by ρ~(r)+r), and ρ~(r)2M+2 for all r2.

In the following, if γ(-1,0), then we will assume first that

r1+γ(2M+2)-γ1,

and will deal with the other situation in the end of this proof. Under the above assumption, we have that r2+γ(ρ~(2r))-γr2+γ(2M+2)-γr when γ<0, and r2+γ(ρ~(2r))-γr2+γ(ρ~(r))-γr2r when γ0. Thus, for all γ>-1 we have

r2+γ(ρ~(2r))-γr.

We will deal with the situation that γ(-1,0) and r1+γ(2M+2)-γ1 at the very end of the proof.

Case 1: r2+γ(ρ~(2r))-γ1+t. If |y-x|r2 and |s-t|r2+γ(ρ~(2r))-γ/4, then we do a scaling:

v(z,τ)=u(rz+x,r2+γ(ρ~(2r))-γτ+t)-u(x0,t)ρ~(2r).

Then

vτ=(|v|2+ε2r2ρ~(2r)-2)γ2(δij+(p-2)vivj|u|2+ε2r2ρ~(2r)-2)uijin Q1.

Notice that εr/ρ~(2r)εr/ρ~(r)ε<1 and r2+γ(ρ~(2r))-γr. Thus, |v(z,τ)|1 for (z,τ)Q1. By applying Corollary 2.4 and Lemma 3.1 to v and rescaling to u, there exists α>0 depending only on γ such that v is Cα in (x,t), and there exists C>0 depending only on n, γ and p such that

|u(y,s)-u(x,s)|Cρ~(2r)|x-y|αrα

and

|u(x,t)-u(x,s)|Cρ~(2r)1+αγ|t-s|αrα(2+γ),

Therefore,

|u(y,s)-u(x,t)|Cρ~(2r)|x-y|αrα+Cρ~(2r)1+αγ|t-s|αrα(2+γ).

Since |y-x|r2 and |s-t|r2+γ(ρ~(2r))-γ/4r4, we have 2-m-1r<|x-y||t-s|2-mr for some integer m1. Then

|u(y,s)-u(x,t)|Cρ~(2m+2(|x-y||t-s|))2mα+Cρ~(2m+2(|x-y||t-s|))1+αγ2mαrα(1+γ)Cρ~(2m+2(|x-y||t-s|))+ρ~(2m+2(|x-y||t-s|))1+αγ2mα.

Notice that

supm1ρ~(2m+2r)+ρ~(2m+2r)1+αγ2mα0as r0.

Therefore, we can choose a modulus of continuity ρ1 such that

ρ1(r)Csupm1ρ~(2m+2r)+ρ~(2m+2r)1+αγ2mαfor all r0,

and we have

|u(y,s)-u(x,t)|ρ1(|x-y||t-s|).

If |y-x|r2, then

|u(x,t)-u(y,s)||u(x,t)-u(x0,t)|+|u(x0,t)-u(y,s)|ρ~(r)+ρ~(|x0-y||t-s|)ρ~(2(|x-y||t-s|))+ρ~((|x-y|+r)|t-s|)ρ~(2(|x-y||t-s|))+ρ~(3(|x-y||t-s|))2ρ~(3(|x-y||t-s|)).

If |x-y|r2 and |s-t|r2+γ(ρ~(2r))-γ/4, then

r412+γ(2M+2)γ2+γ|s-t|12+γ

when γ0, and r2|s-t|1/2 when γ0. Then one can show similar to the above that

|u(x,t)-u(y,s)|2ρ~(c(|x-y||t-s|12|s-t|12+γ))ρ2(|x-y||t-s|),

where ρ2(r)=2ρ~(cr1/2) or ρ2(r)=2ρ~(cr1/(2+γ)) depending on whether γ0 or γ0 is a modulus of continuity, c is a positive constant depending only on M and γ.

This finishes the proof in this first case.

Case 2: r2+γ(ρ~(2r))-γ1+t.

Then let λ=|t+1| when γ0, and λ=(2M+2)γ/(2+γ)|t+1|1/(2+γ) when γ(-1,0). Then one can check that λr.

If |y-x|λ2 and |s-t|λ2+γ(ρ~(2λ))-γ/4, let

v(z,τ)=u(λz+x,λ2+γ(ρ~(2λ))-γτ+t)-u(x0,t)ρ~(2λ)for (z,τ)Q1.

Then

vτ=(|v|2+ε2r2ρ~(2λ)-2)γ2(δij+(p-2)vivj|u|2+ε2λ2ρ~(2λ)-2)uijin Q1.

Notice that λ2+γ(ρ~(2λ))-γλ2λ when γ0, and λ2+γ(ρ~(2λ))-γλr1+γ(ρ~(2r))-γλ when γ(-1,0). Thus, |v(z,τ)|1 for (z,τ)Q1. Also, ελ/ρ~(2λ)ελ/ρ~(λ)ε<1. Then, by arguments similar to the ones in case 1, we have

|u(y,s)-u(x,t)|ρ1(|x-y||t-s|).

If |y-x|λ2, then |t+1|c(|x-y|2|x-y|2+γ)c|x-y| for some c>0 depending only on M and γ. Therefore,

|u(x,t)-u(y,s)||u(x,t)-u(x,-1)|+|u(x,-1)-u(y,s)|ρ~(|t+1|)+ρ~(|x-y||1+s|)ρ~(c|x-y|)+ρ~((|x-y|)|1+t|)ρ~(c(|x-y||t-s|))+ρ~(c|x-y||t-s|)=2ρ~(c(|x-y||t-s|))ρ2(|x-y||t-s|).

If |x-y|λ2 and |s-t|λ2+γ(ρ~(2λ))-γ/4, then

λ412+γ(2M+2)γ2+γ|s-t|12+γ

when γ0, and λ2|s-t|1/2 when γ0. Then one can show similar to the above that

|u(x,t)-u(y,s)||u(x,t)-u(x,-1)|+|u(x,-1)-u(y,s)|ρ~(|t+1|)+ρ~(|x-y||1+s|)ρ~(c(|s-t|22+γ|s-t|2+γ2))+ρ~((|x-y|)|1+t|)ρ~(c(|s-t|12+γ|s-t|12))+ρ~(c(|s-t|12+γ|s-t|12))ρ2(|x-y||t-s|).

This finishes the proof in this second case.

In the end, we deal with the situation that γ(-1,0) and r1+γ(2M+2)-γ1. Then we have rc for c=(2M+2)γ/(1+γ). Let

λ=(2M+2)γ2+γ|t+1|12+γ.

There exists μ>0 depending only on M and γ such that if |t+1|μ, then λc, c2+γ(ρ~(2c))-γ1+t and λ1+γ(2M+2)-γ1. Then, for t-1+μ, the same arguments as in case 2 work without any change.

Now the final case left is that (x,t)B¯1-c×[-1+μ,0]. Then we only need to consider that

(y,s)B1-c/2×[-1+μ2,0].

It follows from Corollary 2.4 and Lemma 3.1 that there exists a modulus of continuity ρ¯ depending only on n, γ, p and M such that

|u(x,t)-u(y,s)|ρ¯(|x-y||t-s|).

This finishes the final situation.

Then ρ*(r):=ρ1(r)+ρ2(r)+ρ¯(r) is a desired modulus of continuity. The proof of this theorem is thereby completed. ∎

Acknowledgements

Part of this work was done when the second author was visiting California Institute of Technology as an Orr foundation Caltech-HKUST Visiting Scholar. He would like to thank Professor Thomas Y. Hou for the kind hosting and discussions.

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About the article

Received: 2016-09-09

Accepted: 2017-08-17

Published Online: 2017-09-06


Funding Source: Research Grants Council, University Grants Committee

Award identifier / Grant number: ECS 26300716

Funding Source: National Science Foundation

Award identifier / Grant number: DMS-1254332

Award identifier / Grant number: DMS-1362525

The second author was supported in part by Hong Kong RGC grant ECS 26300716. The third author was supported in part by NSF grants DMS-1254332 and DMS-1362525.


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 845–867, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0197.

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