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BY 4.0 license Open Access Published online by De Gruyter April 20, 2022

Existence of weak solutions to a general class of diffusive shallow medium type equations

Nicolas Dietrich
From the journal Forum Mathematicum

Abstract

In this article, we prove existence of weak solutions of a general class of diffusive shallow medium type equations. We truncate the coefficients from above and below. Then we prove existence to the problems associated with the truncated vector fields. At last, we show that the approximating solutions converge to the solution of the initial value problem.

MSC 2010: 35D30; 35K20; 35K65

1 Introduction

Let Ω n be an open bounded set and let ( 0 , T ) with T ( 0 , ) be a time interval. Further, we denote the space time cylinder by Ω T := Ω × ( 0 , T ) . Before we discuss the more general case for coefficients, we first consider the prototype case of a diffusive shallow medium equation given by

(1.1) t u - div ( ( u - z ) α | u | p - 2 u ) = f in  Ω T ,

where the regularity of the functions z : Ω and f : Ω T is discussed in (2.2). In this paper, we treat the equations for the exponents p ( 1 , ) and α ( 0 , ) . Concerning the dimension n , we assume n 2 . However, in physical applications the case n = 2 is the most relevant one. We search for solutions u z in order for the term ( u - z ) α to be well defined. This means that z can be interpreted as an obstacle. The equation is doubly nonlinear since it is nonlinear with respect to both the solution u itself and its gradient. The existence of weak solutions for the Cauchy–Dirichlet problem associated with the prototype equation (1.1) has already been proved in [3]. Further, Hölder continuity for bounded weak solutions has been shown in the case z = 0 and f = 0 in [18, 13]. For the case α + p > 2 , the existence of a weak solution has been established in [1]. In the case α = 0 and z = 0 , equation (1.1) is the parabolic p-Laplace equation. When p = 2 and z = 0 , we get the porous medium equation after we formally apply the chain rule. Moreover, existence results for doubly nonlinear equations with general structure conditions can be found in [2, 6, 10, 17, 7, 12]. Furthermore, existence for doubly nonlinear equations similar to our case has been established in [8]. It is also worth noting that local boundedness of weak solutions to (1.1) for the slow diffusion case α + p > 2 , p < 2 and sufficiently regular f and z has been treated in [15]. Concerning regularity, local Hölder continuity of weak solutions to (1.1) was proved in [16] for p > 2 .

Equations of type (1.1) are used to describe the dynamics of ice or water, where the function z can be interpreted as the elevation of the ground on top of which the water or ice is moving. The function u is the height of the water which is measured with respect to some given ground level. We can further interpret the right-hand side f as snow or rain, falling on the ice or water, respectively. Thus, the assumption u z also makes sense from a physical point of view, since the water or ice always moves above the ground.

In this article, we are concerned with the more general equation

(1.2) t u - div A ( x , t , u , u ) = f in  Ω T := Ω × ( 0 , T ) ,

where A : Ω T × × n n is a vector field which satisfies certain growth and monotonicity conditions given in (2.3) and (2.4) below. We are interested in the existence of weak solutions of the Cauchy–Dirichlet problem associated with (1.2). Already in the prototype case (1.1), the term ( u - z ) α in the diffusive part of the equation leads to problems when we want to find a proper definition of weak solutions to (1.1). Hence, we reformulate the Cauchy–Dirichlet problem in terms of the transformation v := u - z and obtain a more natural definition of weak solutions; see Section 2 for more details. The vector field B arising in the diffusion part of the transformed equation will be dependent on v β , where β is some exponent discussed in Section 2. A similar approach of establishing the definition of weak solutions in terms of the transformation v = u - z can be found in [15, 16, 3] for the model case.

To prove the existence of weak solutions to the Cauchy–Dirichlet problem associated with (1.2), we approximate the transformed problem by truncated problems and show that the solutions of these problems converge to the solution of the original problem. To be more precise, we truncate the vector field B with respect to v from above and below and formulate approximating Cauchy–Dirichlet problems to the truncated vector fields B k . The growth and monotonicity properties of the vector fields allow us to apply Galerkin’s method and prove the existence of weak solutions v k for the approximating problems. As we saw above, the transformation v = u - z leads to the appearance of the gradient of v in the transformed vector field B with respect to some power β instead of just v itself. This turns out to be a delicate issue when we want to justify the pointwise convergence of the approximating solutions. We proceed by establishing a uniform L p -bound for the gradients of v k β and using the reflexivity of the vector space to extract a weakly convergent subsequence of ( v k β ) . Applying a compactness argument as in [3, Corollary 4.6], we then obtain pointwise convergence of ( v k β ) and further show that the weak and the pointwise limit coincide. At last, we observe pointwise convergence of the gradients v k β and can finally pass to the limit in the weak form of the approximating equations.

The main issue in proving existence for the more general case (1.2) is that we can not use the explicit form of the vector field in the diffusive part of (1.1). This will mainly lead to difficulties when proving existence of the approximating solutions and when establishing that the limit function satisfies the weak form of the original problem.

2 Setting and main theorem

In this section, we discuss the setting in which we interpret our partial differential equation introduced in (1.2). We formulate the Cauchy–Dirichlet problem

(2.1) { t u - div ( A ( x , t , u , u ) ) = f in  Ω T , u = z on  Ω × ( 0 , T ) , u ( , 0 ) = ψ in  Ω ¯ ,

for u : Ω T with u z . For the given functions f, z and ψ, we assume the following integrability and regularity conditions:

(2.2) f L σ p ( Ω T ; 0 ) , z W 1 , σ β p ( Ω ) , 𝚿 := ψ - z L ( Ω ; 0 ) ,

where σ > n + p p and β := 1 + α p - 1 > 1 . Further, p = p p - 1 denotes the Hölder conjugate of p. The vector field A : Ω T × × n n is a Carathéodory function. More precisely, A ( x , t , u , ξ ) is measurable in ( x , t ) and continuous in ( u , ξ ) for almost every ( x , t ) Ω T . Moreover, the vector field A satisfies the following growth and monotonicity conditions:

(2.3) | A ( x , t , u , ξ ) | C | u - z | α | ξ | p - 1 ,
(2.4) ( A ( x , t , u , ξ ) - A ( x , t , u , η ) ) ( ξ - η ) ν | u - z | α ( | ξ | 2 + | η | 2 ) p - 2 2 | ξ - η | 2

for almost every ( x , t ) Ω T and every u , ξ , η n if p [ 2 , ) , and every u , ξ , η n with ξ η if 1 p < 2 and some constants 0 < ν C < . We motivate a natural weak definition for the Cauchy–Dirichlet problem (2.1) in terms of the transformation v = u - z . By formally applying the chain rule as in [15], we can write equation (1.2) in the form

t v - div ( A ( x , t , v + z , β - 1 v β v 1 - β χ { v 0 } + z ) ) = f .

Then we define the new vector field

(2.5) B ( x , t , v , ξ ) := A ( x , t , v + z , β - 1 v 1 - β χ { v 0 } ξ + z )

for ( x , t ) Ω T , v and ξ n . We arrive at the following reformulation of (2.1) in terms of the transformation v = u - z and the vector field B:

(2.6) { t v - div ( B ( x , t , v , v β ) ) = f in  Ω T , v = 0 on  Ω × ( 0 , T ) , v ( , 0 ) = ψ - z in  Ω ¯ .

In Lemma 3.1 below, we establish the growth and monotonicity conditions satisfied by B. In the following, we give the definition of weak solutions of the Cauchy–Dirichlet problem (2.6). This definition can be motivated by multiplying (2.6) by a smooth function with compact support and then integrating formally by parts.

Definition 2.1 (Weak solutions).

We assume that f, z and ψ satisfy the regularity and integrability conditions given in (2.2). Then a function u : Ω T is a weak solution to (1.2) if and only if v := u - z 0 , v β L p ( 0 , T ; W 1 , p ( Ω ) ) and

(2.7) Ω T [ B ( x , t , v , v β ) φ - v t φ ] 𝑑 x 𝑑 t = Ω T f φ 𝑑 x 𝑑 t

for all φ C 0 ( Ω T ) . Further, if this holds true, v is a solution to the first line of (2.6). If the additional conditions v C ( [ 0 , T ] ; L β + 1 ( Ω ) ) and v β L p ( 0 , T ; W 0 1 , p ( Ω ) ) hold and in terms of the initial datum we have v ( 0 ) = ψ - z , then we call v a weak solution to the Cauchy–Dirichlet problem (2.6), and therefore u is a weak solution to (2.1).

For the finiteness of the integrals in (2.7), we refer to Remark 3.2. Our goal is to prove the following theorem.

Theorem 2.2.

If the functions f, z and ψ satisfy (2.2), then the Cauchy–Dirichlet problem (2.1) has a solution in the sense of Definition 2.1.

3 Preliminaries

3.1 Notation

With B ϱ ( y 0 ) we denote the open ball in m , m , with radius ϱ at the center y 0 m . For v , w 0 , we define

𝔟 [ v , w ] := 1 β + 1 ( v β + 1 - w β + 1 ) - w β ( v - w )
= β β + 1 ( w β + 1 - v β + 1 ) - v ( w β - v β ) ,

where β is introduced in Section 2. In the following, we will often use the notation f ( t ) = f ( , t ) for t [ 0 , T ] . By c we denote a generic constant which can change from line to line.

3.2 Properties of B

Using the growth and monotonicity properties of the vector field A, we can verify similar properties for B, given in the following lemma.

Lemma 3.1 (Properties of B).

Let B be as in (2.5). Then B is bounded and monotone, i.e.

| B ( x , t , v , ξ ) | C β 1 - p | ξ + β v β - 1 z | p - 1

for a.e. ( x , t ) Ω T and every v R , ξ R n , and

[ B ( v , ξ ) - B ( v , η ) ] ( ξ - η ) ν β 1 - p χ { v 0 } ( | ξ + β v β - 1 z | 2 + | η + β v β - 1 z | 2 ) p - 2 2 | ξ - η | 2

for a.e ( x , t ) Ω T , all v R 0 and ξ , η R n .

Proof.

At first, we prove the boundedness. For arbitrary v and ξ n , we have, together with the boundedness (2.3), that

| B ( v , ξ ) | = | A ( v + z , β - 1 v 1 - β χ { v 0 } ξ + z ) |
C | v | α | β - 1 v 1 - β χ { v 0 } ξ + z | p - 1
= C β 1 - p | ξ + β v β - 1 z | p - 1 .

Now we prove the monotonicity. For arbitrary v 0 and ξ , η n , we obtain, applying (2.4), that

[ B ( v , ξ ) - B ( v , η ) ] ( ξ - η ) = [ A ( v + z , β - 1 v 1 - β χ { v 0 } ξ + z ) - A ( v + z , β - 1 v 1 - β χ { v 0 } η + z ) ]
β v β - 1 [ β - 1 v 1 - β χ { v 0 } ξ + z - ( β - 1 v 1 - β χ { v 0 } η + z ) ]
ν β 1 - p χ { v 0 } ( | ξ + β v β - 1 z | 2 + | η + β v β - 1 z | 2 ) p - 2 2 | ξ - η | 2 .

The proof is finished. ∎

Now we show that the integral on the left-hand side of (2.7) is finite.

Remark 3.2.

For v, v β and φ as in Definition 2.1, we have

Ω T | B ( v , v β ) φ | 𝑑 x 𝑑 t < .

Proof.

Applying the boundedness of B given in Lemma 3.1, Hölder’s inequality and Young’s inequality, we observe that

Ω T | B ( v , v β ) φ | 𝑑 x 𝑑 t C β 1 - p sup Ω T | φ | Ω T | v β + β v β - 1 z | p - 1 𝑑 x 𝑑 t
c [ Ω T | v β | p - 1 𝑑 x 𝑑 t + Ω T | v | ( β - 1 ) ( p - 1 ) | z | p - 1 𝑑 x 𝑑 t ]
c ( Ω T , p , φ ) [ v β L p ( Ω T ) p - 1 + v β L p ( Ω T ) p + z L p ( Ω ) p β ( p - 1 ) p + β - 1 ] .

Since v β L p ( 0 , T ; W 1 , p ( Ω ) ) and z W 1 , σ β p ( Ω ) , the right-hand side is finite. ∎

3.3 Mollification in time

Because not every function in L 1 ( Ω T ) is differentiable in time, we define a mollification in time. These mollifications are called Steklov means.

Definition 3.3.

Let f L 1 ( Ω T ) and h ( 0 , T ) . Then we define the Steklov mean of f by

[ f ] h ( x , t ) := 1 h t t + h f ( x , s ) 𝑑 s for  ( x , t ) Ω × ( 0 , T - h ) ,

and the reversed Steklov mean of f by

[ f ] h ¯ ( x , t ) := 1 h t - h t f ( x , s ) 𝑑 s for  ( x , t ) Ω × ( h , T ) .

Throughout this paper, we sometimes need the so-called exponential time mollification. For f L 1 ( Ω T ) and h ( 0 , T ) , we set

f h ( x , t ) := 1 h 0 t e s - t h f ( x , s ) d s

for x Ω and t [ 0 , T ] . Moreover, we define the reversed exponential time mollification by

f h ¯ ( x , t ) := 1 h t T e t - s h f ( x , s ) d s .

The following properties for the mollification in time can be found in [11, Lemma 2.2], [5, Lemma 2.2] and [17, Lemma 2.9].

Lemma 3.4.

Let f L 1 ( Ω T ) and p [ 1 , ) . Then the mollification in time f h given in Definition 3.3 has the following properties:

  1. If f L p ( Ω T ) , then f h L p ( Ω T ) . Further,

    f h L p ( Ω T ) f L p ( Ω T ) ,

    and f h f in L p ( Ω T ) . A similar statement holds for f h ¯ .

  2. If f L p ( Ω T ) , then the weak time derivatives of f h and f h ¯ belong to L p ( Ω T ) . The derivatives are given by

    t f h = 1 h ( f - f h ) , t f h ¯ = 1 h ( f h ¯ - f ) .

  3. Let f L p ( 0 , T ; W 1 , p ( Ω ) ) . Then f h L p ( 0 , T ; W 1 , p ( Ω ) ) and f h f in L p ( 0 , T ; W 1 , p ( Ω ) ) as h 0 . Further, if f L p ( 0 , T ; W 0 1 , p ( Ω ) ) , then also f h L p ( 0 , T ; W 0 1 , p ( Ω ) ) . Similar statements are true for f h ¯ .

  4. If f L p ( 0 , T ; L p ( Ω ) ) , then f h , f h ¯ C ( [ 0 , T ] ; L p ( Ω ) ) .

3.4 Useful lemmas

The following lemma states some properties of 𝔟 . The proof can be found in [4, Lemma 2.2 and Lemma 2.3].

Lemma 3.5.

Let v , w 0 and β > 1 . Then there exists a constant c 1 depending only on β such that

(3.1) 1 c | w β + 1 2 - v β + 1 2 | 2 𝔟 [ v , w ] c | w β + 1 2 - v β + 1 2 | 2 ,
(3.2) 𝔟 [ v , w ] 1 c | v - w | β + 1 ,
(3.3) 𝔟 [ v , w ] c | v β - w β | β + 1 β .

Next we state the following parabolic Gagliardo–Nirenberg inequality. The proof is given in [9, Chapter I, Proposition 3.1].

Lemma 3.6.

Let Ω R n and T > 0 . Suppose r > 0 and p > 1 . Then for every

f L ( 0 , T ; L q ( Ω ) ) L p ( 0 , T ; W 0 1 , p ( Ω ) )

we have

Ω T | f | p ( 1 + q n ) 𝑑 x 𝑑 t c [ ess sup t ( 0 , T ) Ω × { t } | f | r 𝑑 x ] p n Ω T | f | p 𝑑 x 𝑑 t

for a constant c = c ( n , p , r , Ω ) .

The next lemma is similar to [3, Lemma 3.10].

Lemma 3.7.

Let v L β + 1 ( Ω T ; R 0 ) such that it satisfies v β L p ( 0 , T ; W 0 1 , p ( Ω ) ) and (2.7). Then we have that

(3.4) Ω T ζ 𝔟 [ v , w ] 𝑑 x 𝑑 t = Ω T ζ [ t w β ( v - w ) + B ( v , v β ) ( v β - w β ) - f ( v β - w β ) ] 𝑑 x 𝑑 t

for all test functions ζ W 1 , ( [ 0 , T ] ; R 0 ) with ζ ( 0 ) = ζ ( T ) = 0 and all w in the space

𝒱 := { w L β + 1 ( Ω T ) : w β L p ( 0 , T ; W 0 1 , p ( Ω ) ) , t w β L β + 1 β ( Ω T ) } ,

Proof.

We insert φ h = ζ ( w β - v β h ) as a test function in (2.7). Note that φ h L p ( 0 , T ; W 0 1 , p ( Ω ) ) . Thus, there exists a sequence

( φ h , j ) j C 0 ( Ω T )

such that φ h , j φ h in L p ( 0 , T ; W 0 1 , p ( Ω ) ) as j . Further, we have that t φ h , j t φ h in L ( β + 1 ) / β . Plugging in φ h j in (2.7) and passing j , we obtain by a density argument that

Ω T B ( v , v β ) φ h d x d t = : I - Ω T v t φ h d x d t = : I I = Ω T f φ h 𝑑 x 𝑑 t .

We take the limit h 0 for I. Applying the boundedness in Lemma 3.1 and Hölder’s and Young’s inequality, we observe

| Ω T ζ B ( v , v β ) v β h d x d t - Ω T ζ B ( v , v β ) v β d x d t |
c ζ W 1 , ( [ 0 , T ] ; 0 ) ( Ω T | v β | p + | v | p ( β - 1 ) | z | p d x d t ) p - 1 p v β h - v β L p ( 0 , T ; W 1 , p ( Ω ) )
0

as h 0 , where in the last line we used Lemma 3.4 (iii). Further, we can apply Young’s inequality to show that the integral in the last line is finite. Therefore,

(3.5) lim h 0 Ω T B ( v , v β ) φ h d x d t = Ω T B ( v , v β ) ( w β - v β ) ζ 𝑑 x 𝑑 t ,

and similarly

(3.6) lim h 0 Ω T f φ h 𝑑 x 𝑑 t = Ω T f ζ ( w β - v β ) 𝑑 x 𝑑 t .

Now we treat the parabolic term II, by applying Lemma 3.4 (iii) and integrating by parts:

Ω T v t φ h d x d t = Ω T v t ( ζ ( w β - v β h ) d x d t
= Ω T ζ v ( w β - v β h ) + ζ v ( t w β - t v β h ) d x d t
Ω T ζ v t w β + ζ β β + 1 v β h β + 1 β + ζ v ( w β - v β h ) d x d t .

Because of (3.5), (3.6) and (2.7), the limit of II exists. Applying integration by parts, we observe that

lim h 0 Ω T v t φ h d x d t Ω T ζ v t w β + ζ ( β β + 1 v β + 1 + v ( w β - v β ) ) d x d t
= Ω T ζ ( v - w ) t w β - ζ 𝔟 [ v , w ] d x d t .

Now we have verified “ ” in (3.4). The reverse inequality follows similarly, using ζ ( w β - v β h ¯ ) as a test function. ∎

We use the next lemma in the proof of our final existence result.

Lemma 3.8.

Every v L β + 1 ( Ω T ; R 0 ) which satisfies v β L p ( 0 , T ; W 0 1 , p ( Ω ) ) and (2.7) has a representative in C ( [ 0 , T ] ; L β + 1 ( Ω ) ) .

Proof.

We suppose ψ C ( ; [ 0 , 1 ] ) such that ψ ( t ) = 1 for t < 1 2 T , ψ ( t ) = 0 for t > 3 4 T , and | ψ | 8 T . For τ ( 0 , 1 2 T ) and ε > 0 so small such that τ + ε < 1 2 T , we define

ξ ε τ ( t ) = { 0 , t < τ , 1 ε ( t - τ ) , t [ τ , τ + ε ] , 1 , t > τ + ε .

We use ζ = ξ ε τ ψ and w = ( v β h ¯ ) 1 / β as test and comparison functions in (3.4). Using (3.3), applying Lebesgue’s differentiation theorem and letting ε 0 , we find that

Ω 𝔟 [ v , w ] ( , τ ) 𝑑 x Ω T | B ( v , v β ) | | v β - v β h ¯ | d x d t = : I h + Ω T | f | | v β h ¯ - v β | d x d t = : I I h + c T Ω T | v β - v β h ¯ | β + 1 β d x d t = : I I I h

for all τ ( 0 , T 2 ) N h where N h has measure zero. Applying Lemma 3.4 (i), Hölder’s inequality, the boundedness in Lemma 3.1 and Young’s inequality, we obtain that I h 0 as h 0 . The integrals I I h and I I I h also tend to zero. We now take a subsequence h j with h j 0 as j , and we set w j := ( v β h j ¯ ) 1 / p and N := j N h j . Because of (3.2) and the continuity of the map w = ( v β h ¯ ) 1 / β , we can find a representative of v which is continuous in [ 0 , T 2 ] . By a reflection argument, we obtain the continuity in [ T 2 , T ] . ∎

4 Approximation argument

For k > 1 , we approximate the vector field B by vector fields B k given by

B k ( x , t , v , ξ ) = B ( x , t , T k ( v ) , β T k ( v ) β - 1 ξ ) = { B ( x , t , 1 k , β 1 k β - 1 ξ ) , v < 1 k , B ( x , t , v , β v β - 1 ξ ) , 1 k v k , B ( x , t , k , β k β - 1 ξ ) , v > k ,

where T k ( s ) = min { k , max { s , 1 k } } .

4.1 Properties of B k

The properties given in the next lemma follow from Lemma 3.1.

Lemma 4.1 (Properties of B k ).

The B k are bounded, i.e.

(4.1) | B k ( x , t , v , ξ ) | C k α | ξ + z | p - 1 C 2 p - 1 k α ( | ξ | p - 1 + | z | p - 1 ) .

The B k are monotone, i.e.

(4.2) [ B k ( x , t , v , ξ ) - B k ( x , t , v , η ) ] ( ξ - η ) ν k - α ( | ξ + z | 2 + | η + z | 2 ) p - 2 2 | ξ - η | 2

for almost all ( x , t ) Ω T and all v R , and ξ , η R n if p [ 2 , ) and ξ , η R n with ξ η if 1 p < 2 . If p < 2 and ξ η - z , then the right-hand side has to be interpreted as zero.

Proof.

At first, we prove the boundedness (4.1). Let 1 k v k . Then, by applying the boundedness of B, we get

| B k ( v , ξ ) | = | B ( v , β v β - 1 ξ ) |
C | v | α | ξ + z | p - 1
C k α | ξ + z | p - 1
C 2 p - 1 k α ( | ξ | p - 1 + | z | p - 1 ) .

The cases v < 1 k and v > k follow similarly. Now we prove the monotonicity (4.2). Again, we only consider the case 1 k v k . The other two cases follow similarly. By applying the monotonicity of the operator B, we observe

[ B k ( v , ξ ) - B k ( v , η ) ] ( ξ - η ) = β - 1 v 1 - β [ B ( v , β v β - 1 ξ ) - B ( v , β v β - 1 η ) ] ( β v β - 1 ξ - β v β - 1 η )
ν v α ( | ξ + z | 2 + | η + z | 2 ) p - 2 2 | ξ - η | 2
ν k - α ( | ξ + z | 2 + | η + z | 2 ) p - 2 2 | ξ - η | 2

for almost every ( x , t ) Ω T and all v and ξ , η n , excluding the case ξ η - z , if p < 2 . ∎

4.2 Weak solutions of the approximating equations

We formulate the approximating equations. Our goal is to find weak solutions for these equations and prove some regularity properties for them.

(4.3) { t v k - div ( B k ( v k , v k ) ) = f - div ( B k ( 0 , 0 ) ) in  Ω T , v k = 1 k on  Ω × ( 0 , T ) , v k ( , 0 ) = 1 k + 𝚿 in  Ω ¯ ,

where 𝚿 = ψ - z . We integrate (4.3) formally by parts and thus obtain the following definition of weak solutions.

Definition 4.2.

Let

v k C ( [ 0 , T ] ; L 2 ( Ω ) ) 1 k + L p ( 0 , T ; W 0 1 , p ( Ω ) ) .

Then v k is a weak solution to the Cauchy–Dirichlet problem (4.3) if and only if

(4.4) Ω T [ B k ( v k , v k ) φ - v k t φ ] 𝑑 x 𝑑 t = Ω T [ f φ + B k ( 0 , 0 ) φ ] 𝑑 x 𝑑 t

for all φ C 0 ( Ω T ) and v k ( , 0 ) = 1 k + 𝚿 in Ω.

Now we prove the existence of a solution to the approximating problem in the sense of Definition 4.2. We will follow the proof of [3, Lemma 5.2]. As there, we proceed by a functional analytic approach making use of Galerkin’s method; cf. Showalter [14, Theorem 4.1, Section III.4].

Lemma 4.3.

Let k > 1 be arbitrary. Then there exists at least one admissible weak solution v k to (4.3) in the sense of Definition 4.2.

Proof.

Suppose k > 1 . We define the shifted vector field

B ~ k ( w , ξ ) := B k ( 1 k + w , ξ )

for w and ξ n . We prove the existence of a function

w C ( [ 0 , T ] ; L 2 ( Ω ) ) L p ( 0 , T ; W 0 1 , p ( Ω ) )

satisfying

(4.5) Ω T [ B ~ k ( w , w ) φ - w t φ ] 𝑑 x 𝑑 t = Ω T [ f φ + B k ( 0 , 0 ) φ ] 𝑑 x 𝑑 t

for all φ C 0 ( Ω T ) and w ( , 0 ) = 𝚿 in Ω. It is easy to see that v k := 1 k + w is a weak solution to (4.3) in the sense of Definition 4.2. We introduce the new vector space V = L 2 ( Ω ) W 0 1 , p ( Ω ) , equipped with the norm V = L 2 ( Ω ) + W 1 . p ( Ω ) , and define : V V by

( w ) , v := Ω B ~ k ( w , w ) v d x , v , w V .

We define F L p ( 0 , T ; V ) by setting

F ( t ) , v := Ω [ f ( , t ) v + B k ( 0 , 0 ) v ] 𝑑 x , v V .

We observe that V L 2 ( Ω ) V . Note that V is dense in L 2 ( Ω ) . We introduce the space

W p ( 0 , T ) = { v L p ( 0 , T ; V ) : v L p ( 0 , T ; V ) } .

Then it is sufficient to prove that there exists a w W p ( 0 , T ) such that

(4.6) 0 T w ( t ) , v ( t ) 𝑑 t + 0 T ( w ( t ) ) , v ( t ) 𝑑 t = 0 T F ( t ) , v ( t ) 𝑑 t

for all v L p ( 0 , T ; V ) , since (4.6) is equivalent to (4.5). We use Galerkin’s method and choose a Schauder basis ( v j ) j of V. Then for each m , we can pick vectors ψ m span ( v 1 , , v m ) = : V m such that ψ m 𝚿 in L 2 ( Ω ) as m . For such an m , the continuity of B ~ k allows us to find an absolutely continuous function w m : [ 0 , T ] V m which solves the following problem:

(4.7) { ( w m ( t ) , v j ) + ( w m ( t ) ) , v j = F ( t ) , v j for  j { 1 , , m }  and a.e.  t , w m ( 0 ) = ψ m ,

on a maximal interval J [ 0 , T ] . Multiplying (4.7) by the component function w m j ( t ) of w m in the basis ( v j ) j = 1 m and summing over j, we have

(4.8) ( w m ( t ) , w m ( t ) ) + ( w m ( t ) ) , w m ( t ) = F ( t ) , w m ( t )

for almost every t J . Applying the boundedness (4.1), we see that B ~ k ( v , - z ) = 0 . Using this fact, the boundedness (4.1), the monotonicity (4.2) of B ~ k and Young’s inequality, we find that

( v ) , v = Ω B ~ k ( v , v ) v d x
= Ω [ B ~ k ( v , v ) - B ~ k ( v , - z ) ] ( v + z ) 𝑑 x - Ω B ~ k ( v , v ) z d x
c 1 Ω | v + z | p 𝑑 x - c 2 [ Ω | v | p - 1 | z | 𝑑 x + Ω | z | p 𝑑 x ]
c 1 Ω | v | p 𝑑 x - c 2 Ω | z | p 𝑑 x
c 1 v W 1 , p ( Ω ) p - c 2 z L p ( Ω ) p ,

where c 1 := c 1 ( k , α , p , Ω ) > 0 and c 2 := c 2 ( k , α , p ) > 0 . The estimate in (4.8), extending F ( t ) with Hahn–Banach’s theorem to an element of W - 1 , p ( Ω ) = ( W 0 1 , p ( Ω ) ) , integrating over ( 0 , T ) and applying Young’s inequality allow us to obtain

(4.9) 1 2 w m ( t ) L 2 ( Ω ) 2 + c 0 t w m ( s ) W 1 , p ( Ω ) p 𝑑 s c z L p ( Ω ) p T + 1 2 ψ m L 2 ( Ω ) 2 + c 0 T F ( t ) W - 1 , p ( Ω ) p 𝑑 s

for all t J . By contradiction, we can show that J = [ 0 , T ] . Moreover, since ψ m 𝚿 as m in L 2 ( Ω ) , the estimate (4.9) shows that ( w m ) m is a bounded sequence in L p ( 0 , T ; V ) . From the definition of , inequality (4.1) and Hölder’s inequality, we see that

( v ) V c v V p - 1 + c z L p ( Ω ) p - 1 ,

with c = c ( p , α , k ) . This implies that ( ( w m ) ) m is a bounded sequence in L p ( 0 , T ; V ) . By reflexivity, we have a subsequence still labeled as ( w m ) m which converges weakly to w L p ( 0 , T ; V ) and for which ( ( w m ) ) m converges weakly to ϱ L p ( 0 , T ; V ) . Further, (4.9) shows that ( w m ( T ) ) m is bounded in L 2 ( Ω ) , so we may assume that ( w m ( T ) ) m converges weakly to some w ~ L 2 ( Ω ) . Now we take φ C ( [ 0 , T ] ) and v V m . Because of (4.7) and integrating over [ 0 , T ] , we see

- 0 T ( w m ( t ) , v ) φ ( t ) 𝑑 t + 0 T ( w m ( t ) ) , φ ( t ) v 𝑑 t = ( ψ m , v ) φ ( 0 ) - ( w m ( T ) , v ) φ ( T ) + 0 T F ( t ) , φ ( t ) v 𝑑 t .

Due to the weak convergences mentioned above, we obtain, by taking m , that

(4.10) - 0 T ( w ( t ) , v ) φ ( t ) 𝑑 t + 0 T ϱ ( t ) , v φ ( t ) 𝑑 t = ( 𝚿 , v ) φ ( 0 ) - ( w ~ , v ) φ ( T ) + 0 T F ( t ) , φ ( t ) v 𝑑 t

for all v V m 0 and any m 0 . Note that (4.10) also holds for arbitrary v V . This shows, by taking φ C 0 ( 0 , T ) , that

- 0 T ( w ( t ) , v ) φ ( t ) 𝑑 t + 0 T ϱ ( t ) , v φ ( t ) 𝑑 t = 0 T F ( t ) , φ ( t ) v 𝑑 t .

This is equivalent to the following problem: Find w W p ( 0 , T ) such that

(4.11) 0 T w ( t ) , v ( t ) 𝑑 t + 0 T ϱ ( t ) , v ( t ) 𝑑 t = 0 T F ( t ) , v ( t ) 𝑑 t

for all v L p ( 0 , T ; V ) . Since w W p ( 0 , T ) , we have that w C ( [ 0 , T ] ; L 2 ( Ω ) ) ; see [14, Proposition 1.2, Section III.1]. Using the test function

φ ( t ) := { 1 ε ( ε - t ) , t [ 0 , ε ] , 0 , t > ε ,

in (4.10), we observe w ( 0 ) = 𝚿 . Therefore, the desired initial condition is satisfied. It only remains to show that ϱ = ( w ) . Since ϱ is the weak limit of ( ( w m ) ) m , it is sufficient to show that ( ( w m ) ) m converges weakly to ( w ) . In order to prove the weak convergence, we will show the L p -convergence of ( w m ) m to w . By Rellich–Kondrachov’s theorem, we have that W 0 1 , p ( Ω ) L p ( Ω ) , and therefore w m w strongly in L p ( Ω T ) . Thus, we can extract a subsequence, again denoted by ( w m ) m , such that w m w almost everywhere in Ω T . Distinguishing the cases 1 < p < 2 and p 2 and using the monotonicity (4.2), we find that

0 T ( w m ) - ( w ) , w m - w 𝑑 t = : I I I - Ω T [ B ~ k ( w m , w ) - B ~ k ( w , w ) ] ( w m - w ) 𝑑 x 𝑑 t = : I V
= Ω T [ B ~ k ( w m , w m ) - B ~ k ( w , w ) ] ( w m - w ) 𝑑 x 𝑑 t
    - Ω T [ B ~ k ( w m , w ) - B ~ k ( w , w ) ] ( w m - w ) 𝑑 x 𝑑 t
= Ω T [ B ~ k ( w m , w m ) - B ~ k ( w m , w ) ] ( w m - w ) 𝑑 x 𝑑 t
c ( ν , k , α ) Ω T { w m w } ( | w m + z | 2 + | w + z | 2 ) p - 2 2 | w m - w | 2 𝑑 x 𝑑 t
c [ Ω T | w m - w | p 𝑑 x 𝑑 t ] γ ,

where γ := max { 1 , 2 p } . Applying (4.8), the weak convergence of w m and ( w m ) , the norm convergence of ψ m and (4.11), one can justify the convergence of III to zero. Now we show that IV converges to zero. Applying Hölder’s inequality and using the fact that ( w m ) m is bounded in L p ( Ω T ) , the dominated convergence theorem, the continuity and the boundedness (4.1) of B ~ k , we observe

0 | Ω T [ B ~ k ( w m , w ) - B ~ k ( w , w ) ] ( w m - w ) 𝑑 x 𝑑 t |
Ω T | B ~ k ( w m w ) - B ~ k ( w , w ) | | w m - w | 𝑑 x 𝑑 t
c [ Ω T | B ~ k ( w m , w ) - B ~ k ( w , w ) | p p - 1 = : D 𝑑 x 𝑑 t ] p - 1 p 0

as m . In total, we observe that w m w in L p ( Ω T ) , and thus there exists another subsequence, denoted by ( w m ) m