On a comparison principle for Trudinger’s equation

We study the comparison principle for non-negative solutions of the equation ∂ (|v|p−2v) ∂t = div(|∇v|∇v). This equation is related to extremals of Poincaré inequalities in Sobolev spaces. We apply our result to obtain pointwise control of the large time behavior of solutions. AMS Classification 2010: 35K65, 35K55, 35B40, 35A02, 35D30, 35D40


Introduction
Among the so-called doubly non-linear evolutionary equations, Trudinger's equation is distinguished. We shall study its solutions in Ω T = Ω × (0, T ], where Ω is a domain in R n . The equation was originally considered by Trudinger in [22], where a Harnack inequality was studied for a wider class of evolutionary equations. He pointed out that no "intrinsic scaling" is needed for (1.1). The equation has two special features: it is homogeneous and it is not translation invariant, except for p = 2 when it reduces to the Heat Equation.
The first property is, of course, an advantage. While the continuity and some other regularity properties are well understood, the question about uniqueness for the Dirichlet boundary value problem seems to be unsettled (under natural assumptions). A difficulty certainly comes with sign-changing solutions.
We shall prove a comparison principle in Theorem 1 for positive weak supersolutions/subsolutions belonging, by definition, to the Sobolev space L p (0, T ; W 1,p (Ω)). We can allow that one of the functions to be compared has zero lateral boundary values (Corollary 1), which is relevant for a related eigenvalue problem. Furthermore, for Perron's method, a proper comparison principle is a sine qua non. Our result also implies that for p ≥ 2, all nonnegative continuous weak solutions are viscosity solutions in the sense of Crandall, Evans, and Lions in [4]. This is Theorem 3.
The equation has an interesting connection to extremals of Poincaré inequalities in the Sobolev space W 1,p 0 (Ω). These are the minimizers of the Rayleigh quotient (We refer the curious reader to [13] and [14].) Any extremal u = u(x) solves the corresponding Euler-Lagrange Equation div |∇u| p−2 ∇u + λ p |u| p−2 u = 0, (1.3) and v(x, t) = e − λp p−1 t u(x) is a solution to Trudinger's Equation. In [5], the following result is proved: where ∆ p v = div(|∇v| p−2 ∇v), then the limit exists in L p (Ω) and u is a solution of (1.3), possibly identically zero. It also is proved that solutions with extremals as initial data are separable. See [16] for some other related results. We address this question anew in Theorem 2, where we prove a uniqueness result in star-shaped domains. In Corollary 5, we treat the large time behavior in C 1,α domains, not necessarily star-shaped.
To the best of our knowledge, the comparison principles known so far are limited; in [1], a comparison principle is proved under the extra assumption where v is a supersolution and u a subsolution. In [8], a comparison principle is proved for a general class of doubly nonlinear equations. The method there is right; however, for the parameter values yielding Trudinger's Equation, we are unable to verify the validity of the proof in that paper. In [9], a Harnack inequality is proved for strictly positive solutions (Theorem 2.1), and in a subsequent comment it is stated that it is valid also for merely non-negative solutions. A similar result for positive solutions has also been obtained in [7]. Local regularity and regularity up to the boundary has been studied in [17]. The equation and its regularity have also been treated in [10], [11], [19] [20], [21] and [23]. See also [2] for a viscosity approach to the equation.
Plan of the paper. In Section 2, we introduce some notation and define weak solutions. In Section 3, we prove the comparison principle. In Section 4, we establish uniqueness of solutions in star-shaped domains with zero lateral boundary condition. In Section 5, we show that one can compare with extremals of the Poincaré inequality in C 1,α -domains, and we use this to study the large time behavior. Finally, in Section 6, we prove that weak solutions are also viscosity solutions when p ≥ 2.

Preliminaries
We use standard notation. If Ω is a bounded and open set in R n , we use the notation Ω T = Ω × (0, T ), and the parabolic boundary of Ω T is We denote the time derivative of a function v by v t . The positive part of a real quantity a is a + = max{a, 0}. Let us define the weak supersolutions, subsolutions and solutions of the equation Notice that although the functions are non-negative, we shall often write |v| p−2 v in place of v p−1 , the reason being that many auxiliary identities are valid also for sign-changing solutions.

A function is a weak solution if it is both a weak super-and subsolution.
By regularity theory, the weak solutions are locally Hölder continuous, see [9] and [23]. It is likely that the weak super-and subsolutions are semicontinuous (upon a change in a set of Lebesgue measure zero), but since we do not know of any reference we shall simply assume that we study only continuous functions. No doubt, semicontinuity would do.

The comparison principle
The uniqueness of sufficiently smooth solutions that coincide on the parabolic boundary is evident. In particular the time derivative is crucial. Indeed, for two such solutions u 1 and u 2 in C(Ω T ) with the same boundary and initial values on ∂ p Ω T we formally use the test approximates the Heaviside function, to obtain As δ → 0 it follows that Integrating we see that The last integral becomes zero at t 1 = 0. We conclude that u 2 ≤ u 1 and switching the functions we get the desired uniqueness u 2 = u 1 . This simple proof was purely formal. It is not clear how to rescue this reasoning without access to the time derivatives. Therefore we must pay careful attention to the proper regularizations 1 .
We have extracted some parts of our proof below of the Comparison Principle from [8]. Unfortunately, sign-changing solutions are not susceptible of our treatment. In the next Theorem, the majorant can become infinite, if the parameter β = 1.
1 A similar remark can be made for "proofs" of the inequality

Theorem 1.
Let v ≥ 0 be a weak supersolution and u ≥ 0 a weak subsolution We will prove (3.3) with βv replaced byṽ. Adding up (2.1) forṽ and (2.2) yields See Lemma 3.2 in Chapter 3-(i) in [3]. We note that if Integrating τ from −h to 0, we obtain Taking Steklov averages in (3.4) we deduce Since the Steklov average is differentiable with respect to t, we may integrate by parts to obtain Note that G ′ δ (s) = H δ (s) and that H δ (s) is an approximation of the Heaviside function H(s). For 2h < t 1 < t 2 < T − h, we also define the cut-off function where ε > 0 is small.
We now choose the test function 2 In the right-hand side it is straight forward to send ε → 0 and h → 0. We now focus on the left-hand side. As ε → 0 it becomes Here we have used that u and −v are locally bounded (Theorem 5.1 in [9]). We may now let h → 0 to obtain Hence, (3.6) implies where We note that for fixed γ > 0 (definingṽ), Ω δ is compactly contained in Ω × (0, T ). Using that ∇ |u| p−2 u − |ṽ| p−2ṽ = (p−1)|u| p−2 (∇u − ∇ṽ)+(p−1)∇ṽ |u| p−2 − |ṽ| p−2 , and rewriting the right-hand side, (3.7) becomes It is straight forward to see that It now remains to verify that the right-hand side tends to zero as δ → 0. The difficulty is that the available inequality 0 < u p−1 −ṽ p−1 < δ with a very small δ does not necessarily imply that also the quantity u p−2 −ṽ p−2 is of the order O(δ), if it so happens that both u andṽ are very small. It is at this point that the assumption of a lower bound c is crucial.
The elementary inequality Thus we have arrived at sinceṽ > v ≥ c. In the case 1 < p < 2, the above elementary inequality is reversed and a similar reasoning yields Hence we can kill the denominator δ below: where we have also used Young's inequality for the terms involving ∇u and ∇ṽ. Since the integral is convergent and meas(Ω δ ) → 0 it is clear that as δ → 0. Therefore, letting δ → 0 in (3.8), we arrive at Finally, we note that if we instead assume u ≥ c (but not necessarily v ≥ c), thenṽ p−1 < u p−1 <ṽ p−1 + δ in Ω δ .
This means that for δ smaller than inf Ω {u p−1 }, also v is bounded away from zero in Ω δ so that the same argument goes through again.

Corollary 1 (Comparison Principle). If inequality (3.2) is valid on the whole parabolic boundary
Proof. Let β > 1 be arbitrary. As t 1 → 0, the right-hand side of inequality (3.3) approaches zero so that at an arbitrary time t 2 . We conclude that The result follows as β → 1.

Comparison in star-shaped domains
In star-shaped domains, we may prove uniqueness, provided that the lateral boundary values are zero. Convex domains are of this type. The initial values, say g(x), have to be attained for a supersolution (or subsolution) v at least in the sense that In the comparison principle below, at least the "smaller function" has zero lateral boundary values. (We aim at the eigenvalue problem (1.3).) The initial values are taken in the sense of (4.1).

Theorem 2.
Suppose Ω is star-shaped. Let v ≥ 0 be a weak supersolution with initial values v(x, 0) = g(x). Assume that u ≥ 0 is a weak subsolution with the same initial values u(x, 0) = g(x) and with zero lateral values: Then v ≥ u in Ω T . Proof. We may exclude the case v ≡ 0, since then also g ≡ 0. Thus, v > 0 in Ω T by the weak Harnack inequality (Theorem 7.1 in [9]). We may assume that Ω is star-shaped with respect to the origin. Consider the weak supersolution where c α is a positive constant. Hence, the comparison principle in Theorem 1 applies for w α and u in the subdomain Ω × [t 1 , t 2 ] so that when β > 1. Now we can send t 1 to 0+. It follows that We can now safely send α → 1− and β → 1+, whence In the next corollary we assume that the lateral boundary values are taken in the sense that v ∈ L p loc (0, T ; W 1,p 0 (Ω). This is convenient when comparing solutions of the eigenvalue problem (1.3).
The initial values are as in (4.1).

Corollary 2.
Assume g ∈ L p (Ω) and Ω star-shaped. Then the solution v of the following problem In particular, we obtain that solutions with extremals of (1.2) as initial data are separable.
where u p is an extremal of (1.2). Then v = e − λpt p−1 u p .

Extremals and large time behavior
When the initial data are comparable to extremals of (1.2), we are able to extend the comparison principle to more general domains than star-shaped ones. If the boundary is C 1,α -regular, it is known that the extremals are C 1,α up to the boundary and that the estimates are uniform if the C 1,α -norm of the boundary is uniformly controlled. See Theorem 1 in [12]. A consequence of this is the following lemma for the extremals of (1.2) under an exhaustion of the domain.

Lemma 1.
Suppose Ω is C 1,α -domain. Then there are a sequence of uniformly C 1,α -regular sets Ω j exhausting Ω and a sequence of numbers c j → 1 such that c j u j p ≤ u p in Ω j . Here u j p and u p are the extremals of (1.2) in Ω j and Ω, respectively.
Proof. We argue by contradiction. Take ε > 0. If the result is not true, then for any choice of Ω j and each j > 0, there is a point x j ∈ Ω j such that It is clear that x j must converge to a point in ∂Ω since the quotient is uniformly convergent to 1 strictly inside. Suppose for simplicity that x j → 0 ∈ ∂Ω and that near the origin the boundary of Ω is the hyperplane x 1 = 0 3 . We choose the boundary of Ω j to be the hyperplane x 1 = 1/j . By Theorem 1 in [12], the functions u j p are uniformly C 1,α (U ∩ {x 1 ≥ 1/j}) for some neighborhood U of the origin. By inspecting the functions it is easy to conclude from Ascoli's Theorem that up to a subsequence, ∇v j (0) = ∇u j p (e 1 /j) converges to ∇u p (0) when j → ∞.
We may also assume that x j = e 1 /j + z j e 1 where z j > 0 and z j → 0 (otherwise we may just translate the origin along the hyperplane). We note that ∇u p (0) and ∇u j p (x j ) both point in the e 1 direction. Taylor expansion gives Since ∇u j p (e 1 /j) converges to ∇u p (0) we may write ∇u j p (e 1 /j) = |∇u p (0)|e 1 + δ j e 1 , δ j → 0. Again, using Taylor expansion where δ j → 0. Hence since |z j | ≤ |x j |. Therefore, using (5.2) again, This contradicts the antithesis (5.1).
In the next proposition the assumption that g ≥ u p admits multiplication of u p by arbitrarily small constants. Thus the restriction is crucial only near the boundary ∂Ω.
Assume in addition that g ≥ u p where u p is an extremal of (1.2). Then in Ω × (0, T ). Similarly, if g ≤ u p then the reverse inequality holds.
Proof. By Harnack's inequality (Theorem 2.1 in [9]) we can again conclude that v > 0 in Ω T . Indeed, if v vanishes at some point then g vanishes identically, which is excluded by the hypothesis.
Let Ω j be a sequence of smooth subdomains exhausting Ω. Let u j p be the extremal in Ω j with eigenvalue λ j p and the same L p -norm as u p . On page 189 in [14], it is argued for that λ j p → λ p . By Lemma 1 we may choose Ω j such that there are constants c j → 1 such that c j u j p ≤ u p in Ω j for j large enough.
We may pass to the limit in the above inequality and conclude that v ≥ e − λp p−1 t u p . The reversed inequality can be proved similarly.
As a corollary we obtain that solutions with extremals of (1.2) as initial data are separable.
where u p is an extremal of (1.2). Then v = e − λp p−1 t u p .
We can now obtain pointwise control of the large time behavior assuming that the initial data satisfies 0 ≤ g ≤ u p where u p is an extremal of (1.2).
Corollary 5. Assume that Ω is a C 1,α domain and that g ∈ C(Ω) satisfies 0 ≤ g ≤ u p where u p is an extremal of (1.2). Then the convergence is uniform in Ω.
If in addition g ≥ w p where w p is another extremal, then the limit function u(x) is non-zero and therefore an extremal.
Proof. Let τ k be an increasing sequence of positive numbers such that τ k → ∞ as k → ∞. In Theorem 1 in [5] it is proved that lim k→∞ e λpτ k p−1 v(·, τ k ) exists in L p (Ω). We now argue that this convergence is uniform in Ω. Let Remark that e − λpt p−1 u p is a solution. By the comparison with extremals (Proposition 1) and the fact that v is non-negative, for (x, t) ∈ Ω×[−1, 1] for all k ∈ N large enough. These bounds with together with the local Hölder continuity (Theorem 2.8 in [10] and Theorem 2.5 in [11]) give that v k is uniformly bounded in C α (B ×[0, 1]) for any ball B ⊂⊂ Ω. Using a covering argument and that u p is continuous up to the boundary, it is standard to conclude that v k is equicontinuous in Ω × [0, 1]. From this the result follows as in the proof of Theorem 1.3 in [6]. The last part of the statement follows from the observation that in this case we also have the bound from below v k (x, t) ≥ e − λpt p−1 w p (x), which forces the limit to be non-zero. The result then follows from Theorem 1 in [5].
Remark 3. We note that Corollary 5 can be proved using Theorem 2 if Ω is a star-shaped Lipschitz domain and not necessarily a C 1,α domain.

Weak and viscosity solutions
We shall prove that weak solutions are also viscosity solutions when p ≥ 2; see the definition in [2]. As always, this requires at least a comparison principle for classical solutions.
in Ω T is also a viscosity solution.
Proof. We treat supersolutions and subsolutions separately. The result follows by combining the parts.