Existence of solutions to a diffusive shallow medium equation

In this article, we establish the existence of weak solutions to the shallow medium equation. We proceed by an approximation argument. First, we truncate the coefficients of the equation from above and below. Then, we prove convergence of the solutions of the truncated problem to a solution to the original equation.


Introduction
In this work, we prove the existence of weak solutions to the Cauchy-Dirichlet problem associated with the doubly nonlinear equation where ⊂ R n is an open bounded set, z : → R and f : T → R are given sufficiently regular functions, and the parameters α and p are restricted to the range p > 1, and α > 0.
The solution u is required to satisfy u ≥ z so that the factor (u − z) α is well defined. The term doubly nonlinear refers to the fact that the diffusion part depends nonlinearly both on the gradient and on the solution itself. In this article, we consider all n ≥ 2, although the equation is typically used in models with only two spatial dimensions.
In the case z = 0, (1.1) reduces to a standard doubly nonlinear equation which was studied already in [27] and [21], although using a different notion of weak solution. If we furthermore set α = 0, we recover the parabolic p-Laplace equation, whereas taking p = 2 we obtain the porous medium equation, after a formal application of the chain rule. Thus, (1.1) can be seen as a generalization of many previously studied equations.
Local boundedness of weak solutions to (1.1) has been proved in [24] for sufficiently regular f and z in the slow diffusion case α + p > 2. The article focused on the case p < 2, but the value of p does not really play a role in the arguments. In [25], local Hölder continuity was established for p > 2. Existence of solutions to a Cauchy-Dirichlet problem corresponding to (1.1) was proved in the case of trivial topography z = 0 and slow diffusion α + p > 2 in [3]. See also [4,7,15,26] which study the existence of solutions to the Cauchy-Dirichlet problem for doubly nonlinear equations with general structure conditions. Existence, comparison properties and asymptotic behavior of solutions to a nonlinear parabolic problem related to our case have been studied in [8].
In the range p > 2, Eq. (1.1) is used to describe the dynamics of glaciers in the so-called shallow ice approximation; see for example Chapter 2 of [20] and the classic work [19]. Laboratory experiments, theoretical considerations and field measurements suggest that in these applications p ≈ 4 provides the best description. This corresponds to the value 3 for the exponent appearing in the flow law for polycrystalline ice which was established in [13]. In the range p < 2, Eq. (1.1) is used to model shallow water dynamics in situations such as floods and dam breaks; see [3,11,14]. Due to the variety of applications, we follow the terminology from [25] and use the term diffusive shallow medium equation or more concisely, DSM equation, to describe Eq. (1.1) for all p > 1.
The given function z describes the elevation of the land on top of which the water or ice is moving, measured with respect to some arbitrarily chosen ground level. The where β := 1 + α p − 1 > 1. (2.4) Throughout the article, we will focus on this form of the equation. In order to simplify the notation, we denote the vector field appearing in the diffusion part of (2.3) as A(v, ξ ) := β 1− p |ξ + βv β−1 ∇z| p−2 (ξ + βv β−1 ∇z), (2.5) for v ≥ 0 and ξ ∈ R n , so that the Cauchy-Dirichlet problem (2.2) can be rewritten as We arrive at the following definition of weak solutions by multiplying (2.3) by a smooth test function and integrating formally by parts. for all ϕ ∈ C ∞ 0 ( T ). Moreover, in this case we say that v is a solution to (2.3). If in addition v ∈ C([0, T ]; L β+1 ( )), v β ∈ L p (0, T ; W 1 k + . The next crucial step is to investigate the properties of the approximating solutions. We prove the lower bound v k ≥ 1 k and an upper bound for v k independent of k. For large k, this allows us to conclude that Next, we prove a uniform bound for the L p ( T )-norms of the gradients ∇v β k . This shows that there exists a weakly convergent subsequence of v β k in L p (0, T ; W 1, p ( )) to some function w. By a delicate compactness argument (see Sect. 4), we identify the pointwise limit of v k as v := w 1 β . In the next step, we use the differential equations for v k , the ellipticity of the vector field A and the previously proved convergences to conclude pointwise convergence of the gradients ∇v β k → ∇v β for a subsequence. This finally allows us to pass to the limit in the differential equations for v k and in turn to conclude that u = v + z is the desired weak solution to the shallow medium equation.
We remark that there are also other ways of utilizing compactness to obtain the existence. For example, one could apply Corollary 7 of [23] to the sequence (v β+1 k ), similarly as in the proof of Corollary 6.7 in [18]. One then needs to show that ∂ t v β+1 k exists and is bounded in a suitable space, but this can be done by using the mollified formulation (5.9) with a test function involving ([v k ] h ) β . The fact that v k itself has a gradient makes the argument possible. While such an approach is rather elegant, the compactness results that we obtain have the advantage of being directly applicable to similar problems in future research.

Preliminaries
Here, we introduce some notation and present auxiliary tools that will be useful in the course of the paper.

Notation
where β is introduced in (2.4). For convenience, we will sometimes use the shorthand notation w(t) = w(·, t) for t ∈ [0, T ]. By c, we denote a generic constant which may change from line to line.

Mollifications in time
Since weak solutions are not necessarily weakly differentiable with respect to time, we will use some mollification techniques. We first recall the definition of Steklov averages. Given a function v ∈ L 1 ( T ) and h ∈ (0, T ), we define its Steklov mean by (3.2) and the reversed Steklov mean by For some basic properties of Steklov averages, we refer to Chapter 1.3 of [9] and Sect. 9 of [10]. For porous medium-type equations and more generally doubly nonlinear equations, it is often necessary to work with the following so-called exponential time mollification, cf. [17]. For a function v ∈ L 1 ( T ) and h ∈ (0, T ), we set for x ∈ and t ∈ [0, T ]. Moreover, we define the reversed analogue by For details regarding the properties of the exponential mollification, we refer to [ [[v]]h ∈ C([0, T ]; L p ( )). Vol. 21 (2021) Existence of weak solutions to a DSM equation 851

Elementary inequalities
We now recall some elementary inequalities that will be used later, and start by some useful estimates for the quantity b[v, w] that was defined in (3.1). The proof can be found in [ The following algebraic lemma is standard and can be obtained by reasoning as in the proof of Lemma 5.1 in Chapter IX of [9]. Lemma 3.3. Let p > 1. Then, for any ξ, η ∈ R n which in the case p ∈ (1, 2) are not both zero there holds where c is a constant depending only on p. Remark 3.4. Let p > 1 and ξ, η, ζ ∈ R n with either ξ = ζ or η = ζ in the case p ∈ (1, 2). Then, by Lemma 3.3 there holds If p ≥ 2, we use |ξ − η| 2 ≤ 2(|ξ + ζ | 2 + |η + ζ | 2 ) to conclude Lemma 3.5. Let p > 1. Then, for any ξ, η ∈ R n there holds Proof. By Young's inequality, we obtain This is the claimed inequality. Lemma 3.6. Let p > 1 and ξ, η ∈ R n . In the case p < 2, we also assume that either ξ or η is nonzero. Then, there exists a constant c depending only on p such that: which is the desired estimate.

Auxiliary tools
The following lemma can be proven using an inductive argument; see for example [9, § I, Lemma 4.1].
j=0 be a sequence of positive real numbers such that then (Y j ) converges to zero as j → ∞.
Next, we recall a well-known parabolic Sobolev inequality, which can be found for example in [9]. We now present a version of the Cauchy-Peano existence theorem which will be used in connection with Galerkin's method. Proof. Since the differential equation is equivalent to and since A −1 • g is continuous and A −1 •f is integrable, we may assume A = I m×m . Fix r > 0 and set M := maxB r (y o ) |g|. Define δ 1 := r 2M+1 and pick δ 2 > 0 so small that We The set K is convex and closed. Furthermore, the elements of K are equicontinuous pointwise bounded functions, so the Arzelà-Ascoli theorem guarantees that K is compact. We note that we have a well-defined map T : K → K defined by |g(u(s)) − g(v(s))|, the uniform continuity of g on the compact setB r (y o ) shows that T is continuous K → K . Thus, Schauder's fixed point theorem guarantees that there exists at least one y ∈ K such that T y = y. But then y is a solution to the original problem.

Time continuity
We have taken a certain time continuity as part of the definition of a solution to the Cauchy-Dirichlet problem. Our next goal is to show that all weak solutions vanishing on the lateral boundary and having sufficiently high integrability automatically have this type of time continuity. The extra integrability condition v ∈ L β+1 ( T ) occurring in the two following lemmas is redundant in the case α + p ≥ 2, since then βp ≥ β +1. Note that we will only apply the results to the bounded solution found in Sect. 6, which obviously satisfies the extra integrability condition.
The parabolic term is treated as follows: In the third step, we have used Lemma 3.1 (ii) to conclude that the last term is nonpositive, and the second term has been integrated by parts. The limit of the parabolic term as h → 0 exists due to (3.6), (3.7) and (2.7) and satisfies the estimate In the last step, we integrate the second term by parts. Combining (3.6), (3.7) and (3.8), we have verified "≤" in (3.5). The reverse inequality follows in a similar way by taking ϕ = ζ(w β − [[v β ]]h) as a test function.
Using the previous lemma, we can now conclude the time continuity.
Using identity (3.5) from Lemma 3.10 with ζ = χ τ ε ψ and w = ( Here, we used Lemma 3.1 (ii) to conclude that the term involving ∂ t w β is nonpositive. Also, we have made use of Lemma 3.2 (iii) to estimate b[v, w]. Passing to the limit ε → 0, we see that is a set of measure zero. Using Lemma 3.2 (ii), we find a lower bound for the integrand on the left-hand side: Taking now a sequence h j ↓ 0 and setting

Compactness results
In this section, we provide some compactness results which could be of interest in their own. They are consequences of the well-known compactness results in [23]. Such results are necessary when dealing with porous medium-type equations and doubly nonlinear equations. For the purpose of this paper, we only need Corollary 4.6. Since there is not much extra effort to obtain other versions, we preferred to include them as well for possible future applications. Some results in this direction were previously obtained for uniformly bounded functions that are piecewise constant in time; see [16,Lemma 8] and the references therein.
For a family of functions F ⊂ L 1 (0, T ; L 1 ( ; R N )) with N ≥ 1, we denote We start with an Arzéla-Ascoli-type theorem. It will be the basis for all the other compactness results. For a map f : [0, T ] → X , we define the translated function We can now formulate the following theorem [23, § 3, Theorem 1]: and Using Theorem 4.1, we obtain the following result.
Define θ := max{1, mp} and μ := max{1, mq}. Let T > 0 and let ⊂ R n be a bounded domain. Let X be a Banach space with compact embedding X → L q ( ; R N ). Moreover, let and and Before proceeding with the proof, we note that the assumption

Proof of Theorem 4.2. Our aim is to apply Theorem 4.1 to the functions F m with
Otherwise, if m ≥ 1, then θ = mp and μ = mq and we compute This ensures that also (4.2) is satisfied. In the case p = ∞, one must again consider the two ranges for m. The calculations are similar. Therefore, Theorem 4.1 yields the com- This interpolation lemma is from [23, § 8, Lemma 8].
Let X be a Banach space and let q, μ ∈ [1, ∞). Throughout the rest of this section, we will consider compact embeddings T : X → L q ( ; R N ) and S : L μ ( ; R N ) → X that are compatible in the following way. For any v ∈ X for which it happens that the function T v belongs to L μ ( ; R N ) and for any w ∈ L μ ( ; R N ), we have where ·, · denotes the dual pairing of X and X . We need the following more complicated version of Lemma 4.3.
Note that the assumption f m ∈ L p (0, T ; X ) must be understood in the same sense as in Lemma 4.2. In the sequel, we do not explicitly write out the embeddings. It will be clear from the context when they should be present.
Proof. We consider v ∈ L p (0, τ ; X ) ∩ L p (0, τ ; L μ ( ; R N )) (i.e., the composition of v with the embedding T results in a function belonging to L p (0, τ ; L μ ( ; R N ))) and w ∈ L θ (0, τ ; where ·, · denotes the dual pairing of X and X . Hence, we may calculate where in the last line we applied Hölder's inequality. Here, we have omitted the embeddings S and T to simplify the notation. In particular, for some constant c = c(m) ≥ 1. The set t was introduced instead of to avoid dividing by zero in the case m > 1. Now, let η 1 > 0. By Lemma 4.3, there exists M η 1 > 0 such that where c = c(m, p). If m ≤ 1, we therefore immediately conclude that where c = c(m, p). If m > 1, we use Hölder's inequality to estimate I from below and get with the abbreviation F : We have also used the fact that θ = mp when m > 1. Combining both cases and choosing η 1 in a suitable way, we conclude that for any η 2 > 0 there exists M η 2 > 0 such that If q ≤ m+1 m , the asserted inequality follows by an application of Hölder's inequality. In the other case where q > m+1 m , we apply Lemma 4.3 again and find that for η > 0 there exists M η > 0 such that At this point, the claimed inequality follows by choosing η 2 so small that cM η η 2 ≤ η. With these prerequisites at hand, we are able to prove the following more refined version of Theorem 4.2, where assumption (4.5) is weakened in the sense that and F m is bounded in L p (0, T ; X ). (4.10) Proof. Our aim is to apply Theorem 4.1 with B = L q ( ; R N ) and F m instead of F. From (4.10), we know that F m is bounded in L p (t 1 , t 2 ; X ) and hence in L 1 (t 1 , t 2 ; X ) for any 0 < t 1 < t 2 < T . Due to the compact embedding X → L q ( ; R N ), this implies that { for any 0 < t 1 < t 2 < T . Hence, (4.1) is satisfied. Next, we verify assumption (4.2). For η > 0, we denote by M η > 0 the constant from Lemma 4.4. The application of the lemma yields that holds true for any f ∈ F. Assumptions (4.9) and (4.10) ensure the existence of a constant C > 0 such that f m L p (0,T ;X ) ≤ C and f L θ (0,T ;L μ ( ;R N )) ≤ C for any f ∈ F. Hence, for ε > 0 we may choose η > 0 in the preceding inequality small enough such that for a constant C ε > 0 depending on ε, but not on h. If the assumption of assertion (i) holds, then there is h ε such that if 0 < h < h ε , then for all f ∈ F. But this means that for all f ∈ F when h ∈ (0, h ε ). Since ε > 0 was arbitrary, this verifies Assumption (4.2) in Theorem 4.1. Therefore, the application of the theorem yields the compactness of F m in L p (0, T ; L q ( ; R N )) proving assertion (i).
The second assertion (ii) will follow by an interpolation argument. If θ = 1, then the result already follows by (i). Therefore, it is enough to consider the case 1 < θ = mp. We considerp ∈ [1, p). Without loss of generality, we may assume thatθ := mp > 1.
We interpolate Due to Assumption (4.9), f L θ (0,T ;L μ ( ;R N )) is bounded independent of f and therefore we have that and is satisfied, then F m is relatively compact in Lp(0, T ; L q ( ; R N )) for anyp ∈ [1, p).
Proof. We restrict ourselves to the proof of case (i), since (ii) is completely analogous. Let and η ∈ C ∞ 0 ( ) be a nonnegative function with η ≡ 1 in . We now consider the family of functions F η := {η Our goal is to apply Theorem 4.5 with X = W 1, p 0 ( ; R N ) and F η in place of F. Note that due to the parameter ranges, all elements of X are L q -integrable, and the inclusion T : X → L q ( ; R N ) is compact. Similarly, we have a compact inclusionT : X → L μ ( ; R N ). Furthermore, we have an embedding L μ ( ; R N ) → X given by S :=T • J , wherẽ T : (L μ ( ; R N )) → X is the adjoint ofT , and J is the standard isomorphism from L μ ( ; R N ) to (L μ ( ; R N )) . SinceT is compact, Schauder's theorem guarantees that also S is compact. The condition (4.6) follows directly from the definitions of S, T andT . Notice that assumptions (4.11), (4.12) and (i) imply the corresponding assumptions in Theorem 4.5. Thus, all the assumptions of Theorem 4.5 are satisfied, and hence F m η is relatively compact in L p (0, T ; L q ( ; R N )). In particular, this implies that F m is relatively compact in L p (0, T ; L q ( ; R N )). Since was arbitrary, we may conclude the relative compactness of F m in L p (0, T ; L q ( ; R N )) by a diagonal argument.
In the previous proof, we choose X to consist of compactly supported Sobolev functions since the corresponding space of arbitrary Sobolev functions typically is not compactly embedded into L q . Such a result holds only if is sufficiently regular, for example if satisfies a cone condition; see [2,Theorem 6.3]. In that case, the proof is somewhat simpler.
implies assumption (i) of Theorem 4.6 and implies assumption (ii) of Theorem 4.6.

The approximation scheme
For k > 1, we approximate the vector field A defined in (2.5) by vector fields A k given by for v ∈ R and ξ ∈ R n , where the truncation T k : R → R is defined by Exploiting the definition of A, we can express A k as

Properties of A k
Using the properties of the vector field corresponding to the p-Laplace operator, we can verify the following useful basic properties of A k .

Monotonicity
Due to Remark 3.4, we know that there exists a constant c depending only on p such that holds true for any v ∈ R and ξ, η ∈ R n . If p < 2 and ξ ≡ η ≡ −∇z, then the right-hand side has to be interpreted as zero.

Boundedness
For any v ∈ R and ξ ∈ R n , we have

Coercivity
Due to the definition of A k and Lemma 3.5, we have that for any v ∈ R and ξ ∈ R n .

Weak solutions of the approximating equation
In this section, we want to find weak solutions to the approximating problems ⎧ ⎨ where = ψ − z. By formally integrating by parts, we are led to the following definition.
for all 0 ≤ a < b ≤ T − h. In fact, by approximation with smooth functions, one sees that all ϕ ∈ L p (0, T ; W 1, p 0 ( )) ∩ L ∞ ( T ) are admissible in (5.9). An analogous identity holds true for the Steklov averages [ · ]h and h ≤ a < b ≤ T .
We now prove the existence of a solution to the regularized problem in the sense of Definition 5.1. We will follow the functional analytic approach of Showalter [22] making use of Galerkin's method. In fact, one can reason as in the proof of [22, Theorem 4.1, Sect. III.4], despite the somewhat weaker coercivity condition in our case. For the reader's convenience, we present the full argument below. We have opted to avoid the theory of operators of type M used by Showalter, exploiting instead the stronger monotonicity property of the vector field A k .

Proof. We fix k > 1 and consider the modified vector field
for w ∈ R and ξ ∈ R n . We prove the existence of a function w ∈ C([0, T ]; L 2 ( )) ∩ L p (0, T ; W for all ϕ ∈ C ∞ 0 ( T ), and w(·, 0) = in . Then, v k := 1 k + w is an admissible weak solution in the sense of Definition 5.1. We We define F ∈ L p (0, T ; V ) by setting Recall that we have the inclusions V → L 2 ( ) → V with V being dense in L 2 ( ). Then, (5.10) is equivalent to This equation can be understood in the weak sense using the Bochner integral, or equivalently pointwise a.e. Pick a basis (v j ) ∞ j=1 of V , and for each m ∈ N vectors ψ m ∈ span(v 1 , . . . , v m ) =: V m converging to in L 2 ( ), and consider for a fixed m ∈ N the problem of finding a map w m : for j ∈ {1, . . . m} and a.e. t. w m (0) = ψ m (5.11) Here, (·, ·) denotes the inner product in L 2 ( ) and ·, · denotes the dual pairing of V and V . We define g : R m → R m with components The dominated convergence theorem shows that g is continuous,f is evidently integrable and the matrix with components (v i , v j ), i, j ∈ {1, . . . , m} is invertible. Therefore, Lemma 3.9 guarantees that the problem ( (5.12) The coercivity property (5.6) of A k implies the same property forÃ k , from which we obtain for all v ∈ V that with a constant c > 1 depending on α, p, k and . Using this estimate in (5.12) shows that for a.e. t ∈ J . Here, we have extended F(t) to an element of W −1, p ( ) = (W 1, p 0 ( )) using the same formula as before. Applying Young's inequality to the last term, we find that where c > 1 depends on α, p, k and . Integrating the last inequality, we have for all t ∈ J . This shows that w m L 2 ( ) and the component functions w i m stay bounded on J , and from the system of equations we conclude that w m is absolutely continuous on all of J . If J is not the interval [0, T ], then J = [0, b) where b < T . Then, the uniform continuity of w m and the finite dimension of V m show that there is a limit of w m (t) as t ↑ b which allows us to extend w m , thus contradicting maximality. Hence, w m must indeed be defined on all of [0, T ]. Moreover, since ψ m → in L 2 ( ), the estimate (5.13) shows on the one hand that (w m ) is a bounded sequence in L ∞ (0, T ; L 2 ( )) and on the other hand that (w m ) is a bounded sequence in L p (0, T ; W 1, p ( )). Hence, we infer that w m is a bounded sequence in L p (0, T ; V ). From the definition of A and (5.5), we see that with c = c(α, p, k), and hence (A(w m )) is a bounded sequence in L p (0, T ; V ). By reflexivity, we have a subsequence still labeled as (w m ) which converges weakly to w ∈ L p (0, T ; V ) and for which (A(w m )) converges weakly to ξ ∈ L p (0, T ; V ). Furthermore, (5.13) shows that (w m (T )) is bounded in L 2 ( ), so we may assume that (w m (T )) converges weakly to some w * ∈ L 2 ( ). Take now ϕ ∈ C ∞ ([0, T ]) and v ∈ V m . From (5.11), we see that Integrating this identity, we obtain Due to the weak convergences mentioned above, we obtain by taking m → ∞ that for all v ∈ V m o for any m o ∈ N, and by approximation for all v ∈ V . This shows (by taking ϕ ∈ C ∞ 0 (0, T )) that w + ξ = F, (5.15) in L p (0, T ; V ), and thus w ∈ C([0, T ]; L 2 ( )); see also Proposition 1.2 of Sect. III.1 in [22]. Next, we show that w satisfies the right initial condition. Using the test function where the integral on the left-hand side is taken in the Bochner sense of w as an L 2 ( )-valued map. By the density of V in L 2 ( ), this implies The right-hand side converges to zero as ε ↓ 0. On the other hand, since w ∈ C([0, T ]; L 2 ( )), we know that the limit of the integral average appearing on the left-hand side is w(0). Thus, we have confirmed that w(0) = . It only remains to show that ξ = A(w). Since ξ is the weak limit of (A(w m )), it is sufficient to show that (A(w m )) converges weakly to A(w). In order to prove the weak convergence, we first show the L p -convergence of (∇w m ) to ∇w. From the monotonicity condition (5.4) satisfied by A k , it follows that where c = c( p, k) and W m := |∇w m + ∇z| + |∇w + ∇z|.
In the case p < 2, by Hölder's inequality we may estimate The last factor is bounded independently of m since (w m ) is bounded in L p (0, T ; V ). Thus, setting ν = max{1, 2 p } we have in any case that for a constant c independent of m. In the last step, we have used (5.12) integrated over [0, T ]. The weak convergences of (w m ) to w and (A(w m )) to ξ , the norm convergence of (ψ m ), and the weak lower semicontinuity of the norm applied to the term w m (T ) 2 L 2 ( ) then show that lim sup where in the last step we have used (5.15) applied to w and integrated over [0, T ]. Thus, we have obtained the desired L p -convergence of ∇w m to ∇w. To see that the weak convergence of A(w m ) to A(w) follows from this, we use Lemma 3.6 and obtain Hölder's inequality and the L pconvergence of ∇w m show that the last expression converges to zero as m → ∞, so we have confirmed that A(w) = ξ .

Properties of the approximating solutions
In this section, we investigate the properties of the approximating solutions obtained in Lemma 5.2. We prove lower bounds for the solutions v k in terms of k and upper bounds which are independent of k. Moreover, we obtain a uniform bound for the L p ( T )-norms of the gradients ∇v Proof. We use a comparison principle argument. Consider the version of (5.9) with Steklov means The test function is admissible since v k − 1 k ∈ L p (0, T ; W 1, p 0 ( )), and H δ and H δ are bounded. We define G δ as By the properties of the Steklov average and the convexity of G δ , we have Using this estimate and the fact that f ≥ 0 in the above identity, we obtain Taking h ↓ 0, we find that holds true for a.e. 0 < a < b < T . Note that Thus, whenever the second integrand is nonzero we have v k < 1 k and then This shows that the second integral of (5.16) is nonnegative so we can drop it. Thus, we end up with for a.e. 0 < a < b < T . Since v k ∈ C([0, T ]; L 2 ( )) and v k (0) = 1 k + , we obtain, by taking a ↓ 0 that Our goal is to pass to the limit h ↓ 0 in the mollified differential equation, possibly replacing the equality by a suitable estimate. For the elliptic term, we conclude that The convergence of ∇ϕ in the L p -norm can be seen from the chain rule, and the fact that the outer function s → (T k (s) β − M β ) + is piecewise C 1 with bounded derivative. We now introduce the abbreviation Recall that the boundary term b has been introduced in (3.1). This allows us to treat the parabolic term as The limit of the second term vanishes since G(v k )(x, 0) = G( 1 k + (x)) = 0 and M ≥ sup¯ +1 by assumption. The limits exist since G(v k )(x, t)dx is continuous with respect to t. The continuity can be concluded from the fact that G is Lipschitz and v k ∈ C([0, T ]; L 2 ( )). Therefore, we have after passing to the limits h ↓ 0 and ε ↓ 0 that holds true for any τ ∈ (0, T ]. Using the expression (5.3) for A k , the fact that ∇(ṽ β k − M β ) + = 0 a.e. on the sets {(x, t) ∈ T : v k (x, t) ≥ k} and {(x, t) ∈ T : v k (x, t) ≤ M}, the chain rule and Lemma 3.5, we have Moreover, using the definition of G and Lemma 3.2 (i), we can estimate The last term in the integral on the right-hand side of (5.19) can be estimated using the Schwarz inequality, Young's inequality and the fact that k > 1 as The integral over the first term can be included in the first term on the left-hand side of (5.19), and we end up with for any τ ∈ (0, T ] and a constant c = c(β, p). Since (ṽ β k −M β ) + ∈ L p (0, T ; W 1, p 0 ( )), we have by Young's and Poincaré's inequality for any ε ∈ (0, 1) that Due to the Poincaré inequality, the constant c depends on . Choosing ε small enough, we can reabsorb the terms involving |∇(ṽ β − M β ) + | p into the left-hand side. This leads us to In the first term on the right-hand side, we take the supremum over τ ∈ [0, T ], while in the second one we choose τ = T . Proceeding in this way, we end up with inequality (5.18).
We utilize the previous lemma to show that the integral ofṽ βp k is bounded independently of k.
Thus, we have with a constant c = c(β, p, ). This is the desired bound for the L βp -norm ofṽ k . Now, we are ready to prove the boundedness result. Proof. For M ≥ sup¯ + 1, we define the sequences whereṽ k is defined in (5.17). Furthermore, we denote m := β+1 β and A j : Thus, Using Hölder's inequality, Gagliardo Nirenberg's inequality from Lemma 3.8, (5.21) and the energy estimate from Lemma 5.4, we infer that With the abbreviation G := |∇z| βp + f p , we obtain We can estimate the measure of A j+1 by noting that We use this to estimate the second term in the square brackets of (5.22) and the last factor. The remaining instance of |A j+1 | is treated in the same way except that we drop the factor containing M, which is possible since M > 1. In this way, we obtain In view of Corollary 5.5, this shows that where K is defined in (5.20). Thus, we have verified the iterative estimate of Lemma 3.7 with the choices Note that δ > 0 since σ > n+ p p . In order to apply Lemma 3.7, we also need to have which, using the definitions of C and b, is equivalent to for a constantc depending on n, p, β, . We now choose Thus, M is a constant depending only on n, p, β, T , f, z, σ . Since Y j → 0 with this choice, we have that for all k. But this means that for k > L, we have which proves the claim.
The previous lemma together with Lemma 5.3

and the definition of A k show that
for large k. Thus, for large k, we deduce from the differential equation (5.8) that v k satisfies also the equation for any ϕ ∈ C ∞ 0 ( T ). We now show a generalization of this result for test functions which do not necessarily vanish at time zero. This result will be used to verify that once we have concluded the existence of a solution to the original equation, it will also satisfy the correct boundary value.

Lemma 5.7. For sufficiently large k and
Proof. For ϕ as in the statement of the lemma, we apply (5.23) with the test function ζ ε ϕ, where and pass to the limit ε ↓ 0. This is possible since v k ∈ C([0, T ]; L 2 ( )).
Similarly as in Lemma 3.10, we can show that the weak solution v k also satisfies the following modified weak form.  Proof. For δ ∈ (0, T 2 ), we define and choose ζ = ζ δ and the comparison function w = 1 k in the modified weak form (5.25) of the differential equation. Since ∂ t w = 0, the first term on the right-hand side is zero. Our goal now is to pass to the limit δ ↓ 0. Note that ∇w = 0 and thus Using the definition of the vector field A in (2.5), Lemmas 3.5 and 5.6, we obtain Proof. From Lemma 5.9, we know that (v β k − k −β ) is a bounded sequence in the reflexive Banach space L p (0, T ; W 1, p 0 ( )). Therefore, there is a subsequence (k j ) j∈N with k j → ∞ as j → ∞ and a function w ∈ L p (0, T ; W . This implies that also v β k j w weakly in L p (0, T ; W 1, p ( )).
Our next aim is to ensure strong convergence of (v k j ) and thereby to identify the limit function as the pointwise a.e. limit of the subsequence. For this purpose, we let τ ∈ (0, T ). For h ∈ (0, T − τ ) and δ ∈ (0, min{τ, T − τ − h}), we define and consider ϕ ∈ C ∞ 0 ( ). For k > 1, we use the weak formulation (5.23) with the test function ζ δ ϕ and obtain Passing to the limit δ → 0, we see that for any τ ∈ (0, T ) and h ∈ (0, T − τ ) there holds Regarding v k for fixed times as an element of (W 1, p 0 ( )) , and letting ·, · denote the dual pairing of (W 1, p 0 ( )) and W 1, p 0 ( ), we thus have where in the last line we used the bound for v k from Lemma 5.6. By Hölder's inequality, we continue to estimate guarantee that A(v k j , ∇v β ) converges to A(v, ∇v β ) strongly in L p . The quantity ∇v β k − ∇v β stays bounded in L p due to Lemma 5.9, so by Hölder's inequality we conclude that It only remains to control the first term on the right-hand side of (6.3). This term can be rewritten as where [[v β ]] h denotes the exponential time mollification defined in (3.4). Since A(v k , ∇ v β k ) is bounded in L p uniformly in k, we can use Hölder's inequality to estimate the second term on the right-hand side as with a constant c independent of k. In order to treat the first term, we use the modified weak formulation ( By the convergence properties of v k j , we obtain lim sup where in the last line the term involving the time derivative has been omitted, since it is nonpositive by Lemma 3.1 (ii). Combining the last two estimates and joining the previously obtained bounds and convergence properties for I k j -III k j with (6.3), we end up with lim sup By Lemma 3.1 (iii), we see that the right-hand side converges to zero as h → 0. Thus, we have shown that This proves the claim of the lemma, since for any closed subinterval I ⊂ (0, T ) we can choose ζ ∈ C ∞ 0 (0, T ; [0, 1]) such that ξ | I = 1. In this case, χ I ≤ ξ and the result follows. Proof. We fix a test function ϕ as above, and note that it is sufficient to consider functions satisfying |∇ϕ| ≤ 1. Recall that v k j satisfies (5.24) with k = k j and j sufficiently large. The goal is to pass to the limit j → ∞. The limit lim j→∞ T v k j ∂ t ϕ dx dt = T v∂ t ϕ dx dt follows from the L 1 -convergence of v k j to v and the limits lim j→∞ k −α j T |∇z| p−2 ∇z · ∇ϕ dx dt = 0, are trivial (recall that v k (0) = + 1 k ). It remains to treat the elliptic term. For this, we abbreviate and note that for any δ ∈ (0, τ ) we can estimate T |A(v k , ∇v β k ) · ∇ϕ − A(v, ∇v β ) · ∇ϕ| dx dt (6.5) where c = c( p). The terms on the right-hand side can be treated by the dominated convergence theorem. Hence, for k = k j and for all sufficiently large j, we have Taking into account (6.5), (6.6) and (6.8), we have shown that Taking into account all these limits, we have confirmed (6.4).
Now, we are ready to prove the main theorem.