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

Nicolas Dietrich

doi:10.1515/forum-2021-0320

Artikel Open Access

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

Nicolas Dietrich

Veröffentlicht/Copyright: 20. April 2022

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Manuskript einreichen Informationen für Autor*innen Erkunden Sie dieses Fachgebiet

Aus der Zeitschrift Forum Mathematicum Band 34 Heft 4

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.

Keywords: Shallow medium equation; doubly nonlinear equations; existence; coefficients

MSC 2010: 35D30; 35K20; 35K65

1 Introduction

Let Ω⊂ℝn be an open bounded set and let (0,T) with T∈(0,∞) be a time interval. Further, we denote the space time cylinder by ΩT:=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 ⁢ΩT,

where the regularity of the functions z:Ω→ℝ and f:ΩT→ℝ is discussed in (2.2). In this paper, we treat the equations for the exponents p∈(1,∞) and α∈(0,∞). Concerning the dimension n∈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 ⁢ΩT:=Ω×(0,T),

where A:ΩT×ℝ×ℝn→ℝn 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 ∇⁡vβ, 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 Bk. The growth and monotonicity properties of the vector fields allow us to apply Galerkin’s method and prove the existence of weak solutions vk 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 Lp-bound for the gradients of vkβ and using the reflexivity of the vector space to extract a weakly convergent subsequence of (vkβ). Applying a compactness argument as in [3, Corollary 4.6], we then obtain pointwise convergence of (vkβ) and further show that the weak and the pointwise limit coincide. At last, we observe pointwise convergence of the gradients ∇⁡vkβ 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))=fin ⁢ΩT,u=zon ⁢∂⁡Ω×(0,T),u⁢(⋅,0)=ψin ⁢Ω¯,

for u:ΩT→ℝ with u≥z. For the given functions f, z and ψ, we assume the following integrability and regularity conditions:

(2.2)f∈Lσ⁢p′⁢(ΩT;ℝ≥0),z∈W1,σ⁢β⁢p⁢(Ω),𝚿:=ψ-z∈L∞⁢(Ω;ℝ≥0),

where σ>n+pp and β:=1+αp-1>1. Further, p′=pp-1 denotes the Hölder conjugate of p. The vector field A:ΩT×ℝ×ℝn→ℝn 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)∈ΩT. 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-22⁢|ξ-η|2

for almost every (x,t)∈ΩT and every u∈ℝ, ξ,η∈ℝn if p∈[2,∞), and every u∈ℝ, ξ,η∈ℝn 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⁢∇⁡vβ⁢v1-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⁢v1-β⁢χ{v≠0}⁢ξ+∇⁡z)

for (x,t)∈ΩT, v∈ℝ and ξ∈ℝn. 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,∇⁡vβ))=fin ⁢ΩT,v=0on ⁢∂⁡Ω×(0,T),v⁢(⋅,0)=ψ-zin ⁢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:ΩT→ℝ is a weak solution to (1.2) if and only if v:=u-z≥0, vβ∈Lp⁢(0,T;W1,p⁢(Ω)) and

(2.7)∬ΩT[B⁢(x,t,v,∇⁡vβ)⋅∇⁡φ-v⁢∂t⁡φ]⁢𝑑x⁢𝑑t=∬ΩTf⁢φ⁢𝑑x⁢𝑑t

for all φ∈C0∞⁢(ΩT). Further, if this holds true, v is a solution to the first line of (2.6). If the additional conditions v∈C⁢([0,T];Lβ+1⁢(Ω)) and vβ∈Lp⁢(0,T;W01,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 Bϱ⁢(y0) we denote the open ball in ℝm, m∈ℕ, with radius ϱ at the center y0∈ℝm. For v,w≥0, we define

𝔟⁢[v,w]:=1β+1⁢(vβ+1-wβ+1)-wβ⁢(v-w)

=ββ+1⁢(wβ+1-vβ+1)-v⁢(wβ-vβ),

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⁢β1-p⁢|ξ+β⁢vβ-1⁢∇⁡z|p-1

for a.e. (x,t)∈ΩT and every v∈R, ξ∈Rn, and

[B⁢(v,ξ)-B⁢(v,η)]⋅(ξ-η)≥ν⁢β1-p⁢χ{v≠0}⁢(|ξ+β⁢vβ-1⁢∇⁡z|2+|η+β⁢vβ-1⁢∇⁡z|2)p-22⁢|ξ-η|2

for a.e (x,t)∈ΩT, all v∈R≥0 and ξ,η∈Rn.

Proof.

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

|B⁢(v,ξ)|=|A⁢(v+z,β-1⁢v1-β⁢χ{v≠0}⁢ξ+∇⁡z)|

≤C⁢|v|α⁢|β-1⁢v1-β⁢χ{v≠0}⁢ξ+∇⁡z|p-1

=C⁢β1-p⁢|ξ+β⁢vβ-1⁢∇⁡z|p-1.

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

[B⁢(v,ξ)-B⁢(v,η)]⋅(ξ-η)=[A⁢(v+z,β-1⁢v1-β⁢χ{v≠0}⁢ξ+∇⁡z)-A⁢(v+z,β-1⁢v1-β⁢χ{v≠0}⁢η+∇⁡z)]

⋅β⁢vβ-1⁢[β-1⁢v1-β⁢χ{v≠0}⁢ξ+∇⁡z-(β-1⁢v1-β⁢χ{v≠0}⁢η+∇⁡z)]

≥ν⁢β1-p⁢χ{v≠0}⁢(|ξ+β⁢vβ-1⁢∇⁡z|2+|η+β⁢vβ-1⁢∇⁡z|2)p-22⁢|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, vβ and φ as in Definition 2.1, we have

∬ΩT|B⁢(v,∇⁡vβ)⋅∇⁡φ|⁢𝑑x⁢𝑑t<∞.

Proof.

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

∬ΩT|B⁢(v,∇⁡vβ)⋅∇⁡φ|⁢𝑑x⁢𝑑t≤C⁢β1-p⁢supΩT⁡|∇⁡φ|⁢∬ΩT|∇⁡vβ+β⁢vβ-1⁢∇⁡z|p-1⁢𝑑x⁢𝑑t

≤c⁢[∬ΩT|∇⁡vβ|p-1⁢𝑑x⁢𝑑t+∬ΩT|v|(β-1)⁢(p-1)⁢|∇⁡z|p-1⁢𝑑x⁢𝑑t]

≤c⁢(ΩT,p,∇⁡φ)⁢[∥∇⁡vβ∥Lp⁢(ΩT)p-1+∥vβ∥Lp⁢(ΩT)p+∥∇⁡z∥Lp⁢(Ω)p⁢β⁢(p-1)p+β-1].

Since vβ∈Lp⁢(0,T;W1,p⁢(Ω)) and z∈W1,σ⁢β⁢p⁢(Ω), the right-hand side is finite. ∎

3.3 Mollification in time

Because not every function in L1⁢(ΩT) is differentiable in time, we define a mollification in time. These mollifications are called Steklov means.

Definition 3.3.

Let f∈L1⁢(ΩT) and h∈(0,T). Then we define the Steklov mean of f by

[f]h⁢(x,t):=1h⁢∫tt+hf⁢(x,s)⁢𝑑s for ⁢(x,t)∈Ω×(0,T-h),

and the reversed Steklov mean of f by

[f]h¯⁢(x,t):=1h⁢∫t-htf⁢(x,s)⁢𝑑s for ⁢(x,t)∈Ω×(h,T).

Throughout this paper, we sometimes need the so-called exponential time mollification. For f∈L1⁢(ΩT) and h∈(0,T), we set

〚f〛h(x,t):=1h∫0tes-thf(x,s)ds

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

〚f〛h¯(x,t):=1h∫tTet-shf(x,s)ds.

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∈L1⁢(ΩT) and p∈[1,∞). Then the mollification in time 〚f〛h given in Definition 3.3 has the following properties:

If f∈Lp⁢(ΩT), then 〚f〛h∈Lp(ΩT). Further,
∥〚f〛h∥Lp⁢(ΩT)≤∥f∥Lp⁢(ΩT),
and 〚f〛h→f in Lp⁢(ΩT). A similar statement holds for 〚f〛h¯.
If f∈Lp⁢(ΩT), then the weak time derivatives of 〚f〛h and 〚f〛h¯ belong to Lp⁢(ΩT). The derivatives are given by
∂t〚f〛h=1h(f-〚f〛h),∂t〚f〛h¯=1h(〚f〛h¯-f).
Let f∈Lp⁢(0,T;W1,p⁢(Ω)). Then 〚f〛h∈Lp(0,T;W1,p(Ω)) and 〚f〛h→f in Lp⁢(0,T;W1,p⁢(Ω)) as h↓0. Further, if f∈Lp⁢(0,T;W01,p⁢(Ω)), then also 〚f〛h∈Lp(0,T;W01,p(Ω)). Similar statements are true for 〚f〛h¯.
If f∈Lp⁢(0,T;Lp⁢(Ω)), then 〚f〛h, 〚f〛h¯∈C([0,T];Lp(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)1c⁢|wβ+12-vβ+12|2≤𝔟⁢[v,w]≤c⁢|wβ+12-vβ+12|2,

(3.2)𝔟⁢[v,w]≥1c⁢|v-w|β+1,

(3.3)𝔟⁢[v,w]≤c⁢|vβ-wβ|β+1β.

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⊂Rn and T>0. Suppose r>0 and p>1. Then for every

f∈L∞⁢(0,T;Lq⁢(Ω))∩Lp⁢(0,T;W01,p⁢(Ω))

we have

∬ΩT|f|p⁢(1+qn)⁢𝑑x⁢𝑑t≤c⁢[ess⁢supt∈(0,T)⁢∫Ω×{t}|f|r⁢𝑑x]pn⁢∬ΩT|∇⁡f|p⁢𝑑x⁢𝑑t

for a constant c=c⁢(n,p,r,Ω).

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

Lemma 3.7.

Let v∈Lβ+1⁢(ΩT;R≥0) such that it satisfies vβ∈Lp⁢(0,T;W01,p⁢(Ω)) and (2.7). Then we have that

(3.4)∬ΩTζ′⁢𝔟⁢[v,w]⁢𝑑x⁢𝑑t=∬ΩTζ⁢[∂t⁡wβ⁢(v-w)+B⁢(v,∇⁡vβ)⋅(∇⁡vβ-∇⁡wβ)-f⁢(vβ-wβ)]⁢𝑑x⁢𝑑t

for all test functions ζ∈W1,∞⁢([0,T];R≥0) with ζ⁢(0)=ζ⁢(T)=0 and all w in the space

𝒱:={w∈Lβ+1⁢(ΩT):wβ∈Lp⁢(0,T;W01,p⁢(Ω)),∂t⁡wβ∈Lβ+1β⁢(ΩT)},

Proof.

We insert φh=ζ(wβ-〚vβ〛h) as a test function in (2.7). Note that φh∈Lp⁢(0,T;W01,p⁢(Ω)). Thus, there exists a sequence

(φh,j)j∈ℕ⊂C0∞⁢(ΩT)

such that φh,j→φh in Lp⁢(0,T;W01,p⁢(Ω)) as j→∞. Further, we have that ∂t⁡φh,j→∂t⁡φh in L(β+1)/β. Plugging in φhj in (2.7) and passing j→∞, we obtain by a density argument that

∬ΩTB⁢(v,∇⁡vβ)⋅∇⁡φh⁢d⁢x⁢d⁢t⏟=⁣:I-∬ΩTv⁢∂t⁡φh⁢d⁢x⁢d⁢t⏟=⁣:I⁢I=∬ΩTf⁢φh⁢𝑑x⁢𝑑t.

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

|∬ΩTζB(v,∇vβ)⋅∇〚vβ〛hdxdt-∬ΩTζB(v,∇vβ)⋅∇vβdxdt|

≤c∥ζ∥W1,∞⁢([0,T];ℝ≥0)(∬ΩT|∇vβ|p+|v|p⁢(β-1)|∇z|pdxdt)p-1p∥〚vβ〛h-vβ∥Lp⁢(0,T;W1,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)limh↓0⁡∬ΩTB⁢(v,∇⁡vβ)⋅∇⁡φh⁢d⁢x⁢d⁢t=∬ΩTB⁢(v,∇⁡vβ)⋅(∇⁡wβ-∇⁡vβ)⁢ζ⁢𝑑x⁢𝑑t,

and similarly

(3.6)limh↓0⁡∬ΩTf⁢φh⁢𝑑x⁢𝑑t=∬ΩTf⁢ζ⁢(wβ-vβ)⁢𝑑x⁢𝑑t.

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

∬ΩTv∂tφhdxdt=∬ΩTv∂t(ζ(wβ-〚vβ〛h)dxdt

=∬ΩTζ′v(wβ-〚vβ〛h)+ζv(∂twβ-∂t〚vβ〛h)dxdt

≤∬ΩTζv∂twβ+ζ′ββ+1〚vβ〛hβ+1β+ζ′v(wβ-〚vβ〛h)dxdt.

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

limh→0⁡∬ΩTv⁢∂t⁡φh⁢d⁢x⁢d⁢t≤∬ΩTζ⁢v⁢∂t⁡wβ+ζ′⁢(ββ+1⁢vβ+1+v⁢(wβ-vβ))⁢d⁢x⁢d⁢t

=∬ΩTζ⁢(v-w)⁢∂t⁡wβ-ζ′⁢𝔟⁢[v,w]⁢d⁢x⁢d⁢t.

Now we have verified “≤” in (3.4). The reverse inequality follows similarly, using ζ(wβ-〚vβ〛h¯) as a test function. ∎

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

Lemma 3.8.

Every v∈Lβ+1⁢(ΩT;R≥0) which satisfies vβ∈Lp⁢(0,T;W01,p⁢(Ω)) and (2.7) has a representative in C⁢([0,T];Lβ+1⁢(Ω)).

Proof.

We suppose ψ∈C∞⁢(ℝ;[0,1]) such that ψ⁢(t)=1 for t<12⁢T, ψ⁢(t)=0 for t>34⁢T, and |ψ′|≤8T. For τ∈(0,12⁢T) and ε>0 so small such that τ+ε<12⁢T, we define

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

We use ζ=ξετ⁢ψ and w=(〚vβ〛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

∫Ω𝔟⁢[v,w]⁢(⋅,τ)⁢𝑑x≤∬ΩT|B(v,∇vβ)||∇vβ-∇〚vβ〛h¯|dxdt⏟=⁣:Ih+∬ΩT|f||〚vβ〛h¯-vβ|dxdt⏟=⁣:I⁢Ih+cT⁢∬ΩT|vβ-〚vβ〛h¯|β+1βdxdt⏟=⁣:I⁢I⁢Ih

for all τ∈(0,T2)∖Nh where Nh has measure zero. Applying Lemma 3.4 (i), Hölder’s inequality, the boundedness in Lemma 3.1 and Young’s inequality, we obtain that Ih→0 as h→0. The integrals I⁢Ih and I⁢I⁢Ih also tend to zero. We now take a subsequence hj with hj→0 as j→∞, and we set wj:=(〚vβ〛hj¯)1/p and N:=⋃j∈ℕNhj. Because of (3.2) and the continuity of the map w=(〚vβ〛h¯)1/β, we can find a representative of v which is continuous in [0,T2]. By a reflection argument, we obtain the continuity in [T2,T]. ∎

4 Approximation argument

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

Bk⁢(x,t,v,ξ)=B⁢(x,t,Tk⁢(v),β⁢Tk⁢(v)β-1⁢ξ)={B⁢(x,t,1k,β⁢1kβ-1⁢ξ),v<1k,B⁢(x,t,v,β⁢vβ-1⁢ξ),1k≤v≤k,B⁢(x,t,k,β⁢kβ-1⁢ξ),v>k,

where Tk⁢(s)=min⁡{k,max⁡{s,1k}}.

4.1 Properties of Bk

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

Lemma 4.1 (Properties of Bk).

The Bk are bounded, i.e.

(4.1)|Bk⁢(x,t,v,ξ)|≤C⁢kα⁢|ξ+∇⁡z|p-1≤C⁢2p-1⁢kα⁢(|ξ|p-1+|∇⁡z|p-1).

The Bk are monotone, i.e.

(4.2)[Bk⁢(x,t,v,ξ)-Bk⁢(x,t,v,η)]⋅(ξ-η)≥ν⁢k-α⁢(|ξ+∇⁡z|2+|η+∇⁡z|2)p-22⁢|ξ-η|2

for almost all (x,t)∈ΩT and all v∈R, and ξ,η∈Rn if p∈[2,∞) and ξ,η∈Rn 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 1k≤v≤k. Then, by applying the boundedness of B, we get

|Bk⁢(v,ξ)|=|B⁢(v,β⁢vβ-1⁢ξ)|

≤C⁢|v|α⁢|ξ+∇⁡z|p-1

≤C⁢kα⁢|ξ+∇⁡z|p-1

≤C⁢2p-1⁢kα⁢(|ξ|p-1+|∇⁡z|p-1).

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

[Bk⁢(v,ξ)-Bk⁢(v,η)]⋅(ξ-η)=β-1⁢v1-β⁢[B⁢(v,β⁢vβ-1⁢ξ)-B⁢(v,β⁢vβ-1⁢η)]⋅(β⁢vβ-1⁢ξ-β⁢vβ-1⁢η)

≥ν⁢vα⁢(|ξ+∇⁡z|2+|η+∇⁡z|2)p-22⁢|ξ-η|2

≥ν⁢k-α⁢(|ξ+∇⁡z|2+|η+∇⁡z|2)p-22⁢|ξ-η|2

for almost every (x,t)∈ΩT and all v∈ℝ and ξ,η∈ℝn, 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⁡vk-div⁡(Bk⁢(vk,∇⁡vk))=f-div⁡(Bk⁢(0,0))in ⁢ΩT,vk=1kon ⁢∂⁡Ω×(0,T),vk⁢(⋅,0)=1k+𝚿in ⁢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

vk∈C⁢([0,T];L2⁢(Ω))∩1k+Lp⁢(0,T;W01,p⁢(Ω)).

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

(4.4)∬ΩT[Bk⁢(vk,∇⁡vk)⋅∇⁡φ-vk⁢∂t⁡φ]⁢𝑑x⁢𝑑t=∬ΩT[f⁢φ+Bk⁢(0,0)⋅∇⁡φ]⁢𝑑x⁢𝑑t

for all φ∈C0∞⁢(ΩT) and vk⁢(⋅,0)=1k+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 vk to (4.3) in the sense of Definition 4.2.

Proof.

Suppose k>1. We define the shifted vector field

B~k⁢(w,ξ):=Bk⁢(1k+w,ξ)

for w∈ℝ and ξ∈ℝn. We prove the existence of a function

w∈C⁢([0,T];L2⁢(Ω))∩Lp⁢(0,T;W01,p⁢(Ω))

satisfying

(4.5)∬ΩT[B~k⁢(w,∇⁡w)⋅∇⁡φ-w⁢∂t⁡φ]⁢𝑑x⁢𝑑t=∬ΩT[f⁢φ+Bk⁢(0,0)⋅∇⁡φ]⁢𝑑x⁢𝑑t

for all φ∈C0∞⁢(ΩT) and w⁢(⋅,0)=𝚿 in Ω. It is easy to see that vk:=1k+w is a weak solution to (4.3) in the sense of Definition 4.2. We introduce the new vector space V=L2⁢(Ω)∩W01,p⁢(Ω), equipped with the norm ∥⋅∥V=∥⋅∥L2⁢(Ω)+∥⋅∥W1.p⁢(Ω), and define ℬ:V→V′ by

〈ℬ⁢(w),v〉:=∫ΩB~k⁢(w,∇⁡w)⋅∇⁡v⁢d⁢x,v,w∈V.

We define F∈Lp′⁢(0,T;V′) by setting

〈F⁢(t),v〉:=∫Ω[f⁢(⋅,t)⁢v+Bk⁢(0,0)⋅∇⁡v]⁢𝑑x,v∈V.

We observe that V↪L2⁢(Ω)↪V′. Note that V is dense in L2⁢(Ω). We introduce the space

Wp⁢(0,T)={v∈Lp⁢(0,T;V):v′∈Lp′⁢(0,T;V′)}.

Then it is sufficient to prove that there exists a w∈Wp⁢(0,T) such that

(4.6)∫0T〈w′⁢(t),v⁢(t)〉⁢𝑑t+∫0T〈ℬ⁢(w⁢(t)),v⁢(t)〉⁢𝑑t=∫0T〈F⁢(t),v⁢(t)〉⁢𝑑t

for all v∈Lp⁢(0,T;V), since (4.6) is equivalent to (4.5). We use Galerkin’s method and choose a Schauder basis (vj)j∈ℕ of V. Then for each m∈ℕ, we can pick vectors ψm∈span(v1,…,vm)=:Vm such that ψm→𝚿 in L2⁢(Ω) as m→∞. For such an m∈ℕ, the continuity of B~k allows us to find an absolutely continuous function wm:[0,T]→Vm which solves the following problem:

(4.7){(wm′⁢(t),vj)+〈ℬ⁢(wm⁢(t)),vj〉=〈F⁢(t),vj〉for ⁢j∈{1,…,m}⁢ and a.e. ⁢t,wm⁢(0)=ψm,

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

(4.8)(wm′⁢(t),wm⁢(t))+〈ℬ⁢(wm⁢(t)),wm⁢(t)〉=〈F⁢(t),wm⁢(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

〈ℬ⁢(v),v〉=∫ΩB~k⁢(v,∇⁡v)⋅∇⁡v⁢d⁢x

=∫Ω[B~k⁢(v,∇⁡v)-B~k⁢(v,-∇⁡z)]⋅(∇⁡v+∇⁡z)⁢𝑑x-∫ΩB~k⁢(v,∇⁡v)⋅∇⁡z⁢d⁢x

≥c1⁢∫Ω|∇⁡v+∇⁡z|p⁢𝑑x-c2⁢[∫Ω|∇⁡v|p-1⁢|∇⁡z|⁢𝑑x+∫Ω|∇⁡z|p⁢𝑑x]

≥c1⁢∫Ω|∇⁡v|p⁢𝑑x-c2⁢∫Ω|∇⁡z|p⁢𝑑x

≥c1⁢∥v∥W1,p⁢(Ω)p-c2⁢∥∇⁡z∥Lp⁢(Ω)p,

where c1:=c1⁢(k,α,p,Ω)>0 and c2:=c2⁢(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)=(W01,p⁢(Ω))′, integrating over (0,T) and applying Young’s inequality allow us to obtain

(4.9)12⁢∥wm⁢(t)∥L2⁢(Ω)2+c⁢∫0t∥wm⁢(s)∥W1,p⁢(Ω)p⁢𝑑s≤c⁢∥∇⁡z∥Lp⁢(Ω)p⁢T+12⁢∥ψm∥L2⁢(Ω)2+c⁢∫0T∥F⁢(t)∥W-1,p′⁢(Ω)p′⁢𝑑s

for all t∈J. By contradiction, we can show that J=[0,T]. Moreover, since ψm→𝚿 as m→∞ in L2⁢(Ω), the estimate (4.9) shows that (wm)m∈ℕ is a bounded sequence in Lp⁢(0,T;V). From the definition of ℬ, inequality (4.1) and Hölder’s inequality, we see that

∥ℬ⁢(v)∥V′≤c⁢∥v∥Vp-1+c⁢∥∇⁡z∥Lp⁢(Ω)p-1,

with c=c⁢(p,α,k). This implies that (ℬ⁢(wm))m∈ℕ is a bounded sequence in Lp′⁢(0,T;V′). By reflexivity, we have a subsequence still labeled as (wm)m∈ℕ which converges weakly to w∈Lp⁢(0,T;V) and for which (ℬ⁢(wm))m∈ℕ converges weakly to ϱ∈Lp′⁢(0,T;V′). Further, (4.9) shows that (wm⁢(T))m∈ℕ is bounded in L2⁢(Ω), so we may assume that (wm⁢(T))m∈ℕ converges weakly to some w~∈L2⁢(Ω). Now we take φ∈C∞⁢([0,T]) and v∈Vm. Because of (4.7) and integrating over [0,T], we see

-∫0T(wm⁢(t),v)⁢φ′⁢(t)⁢𝑑t+∫0T〈ℬ⁢(wm⁢(t)),φ⁢(t)⁢v〉⁢𝑑t=(ψm,v)⁢φ⁢(0)-(wm⁢(T),v)⁢φ⁢(T)+∫0T〈F⁢(t),φ⁢(t)⁢v〉⁢𝑑t.

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

(4.10)-∫0T(w⁢(t),v)⁢φ′⁢(t)⁢𝑑t+∫0T〈ϱ⁢(t),v〉⁢φ⁢(t)⁢𝑑t=(𝚿,v)⁢φ⁢(0)-(w~,v)⁢φ⁢(T)+∫0T〈F⁢(t),φ⁢(t)⁢v〉⁢𝑑t

for all v∈Vm0 and any m0∈ℕ. Note that (4.10) also holds for arbitrary v∈V. This shows, by taking φ∈C0∞⁢(0,T), that

-∫0T(w⁢(t),v)⁢φ′⁢(t)⁢𝑑t+∫0T〈ϱ⁢(t),v〉⁢φ⁢(t)⁢𝑑t=∫0T〈F⁢(t),φ⁢(t)⁢v〉⁢𝑑t.

This is equivalent to the following problem: Find w∈Wp⁢(0,T) such that

(4.11)∫0T〈w′⁢(t),v⁢(t)〉⁢𝑑t+∫0T〈ϱ⁢(t),v⁢(t)〉⁢𝑑t=∫0T〈F⁢(t),v⁢(t)〉⁢𝑑t

for all v∈Lp⁢(0,T;V). Since w∈Wp⁢(0,T), we have that w∈C⁢([0,T];L2⁢(Ω)); see [14, Proposition 1.2, Section III.1]. Using the test function

φ⁢(t):={1ε⁢(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=ℬ⁢(w). Since ϱ is the weak limit of (ℬ⁢(wm))m∈ℕ, it is sufficient to show that (ℬ⁢(wm))m∈ℕ converges weakly to ℬ⁢(w). In order to prove the weak convergence, we will show the Lp-convergence of (∇⁡wm)m∈ℕ to ∇⁡w. By Rellich–Kondrachov’s theorem, we have that W01,p⁢(Ω)⋐Lp⁢(Ω), and therefore wm→w strongly in Lp⁢(ΩT). Thus, we can extract a subsequence, again denoted by (wm)m∈ℕ, such that wm→w almost everywhere in ΩT. Distinguishing the cases 1<p<2 and p≥2 and using the monotonicity (4.2), we find that

∫0T〈ℬ⁢(wm)-ℬ⁢(w),wm-w〉⁢𝑑t⏟=⁣:I⁢I⁢I-∬ΩT[B~k⁢(wm,∇⁡w)-B~k⁢(w,∇⁡w)]⋅(∇⁡wm-∇⁡w)⁢𝑑x⁢𝑑t⏟=⁣:I⁢V

=∬ΩT[B~k⁢(wm,∇⁡wm)-B~k⁢(w,∇⁡w)]⋅(∇⁡wm-∇⁡w)⁢𝑑x⁢𝑑t

-∬ΩT[B~k⁢(wm,∇⁡w)-B~k⁢(w,∇⁡w)]⋅(∇⁡wm-∇⁡w)⁢𝑑x⁢𝑑t

=∬ΩT[B~k⁢(wm,∇⁡wm)-B~k⁢(wm,∇⁡w)]⋅(∇⁡wm-∇⁡w)⁢𝑑x⁢𝑑t

≥c⁢(ν,k,α)⁢∬ΩT∩{∇wm≠∇w}(|∇⁡wm+∇⁡z|2+|∇⁡w+∇⁡z|2)p-22⁢|∇⁡wm-∇⁡w|2⁢𝑑x⁢𝑑t

≥c⁢[∬ΩT|∇⁡wm-∇⁡w|p⁢𝑑x⁢𝑑t]γ,

where γ:=max⁡{1,2p}. Applying (4.8), the weak convergence of wm and ℬ⁢(wm), the norm convergence of ψm 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 (∇⁡wm)m∈ℕ is bounded in Lp⁢(ΩT), the dominated convergence theorem, the continuity and the boundedness (4.1) of B~k, we observe

0≤|∬ΩT[B~k⁢(wm,∇⁡w)-B~k⁢(w,∇⁡w)]⋅(∇⁡wm-∇⁡w)⁢𝑑x⁢𝑑t|

≤∬ΩT|B~k⁢(wm⁢∇⁡w)-B~k⁢(w,∇⁡w)|⁢|∇⁡wm-∇⁡w|⁢𝑑x⁢𝑑t

≤c⁢[∬ΩT|B~k⁢(wm,∇⁡w)-B~k⁢(w,∇⁡w)|pp-1⏟=⁣:D⁢𝑑x⁢𝑑t]p-1p→0

as m→∞. In total, we observe that ∇⁡wm→∇⁡w in Lp⁢(ΩT), and thus there exists another subsequence, denoted by (∇⁡wm)m∈ℕ, such that ∇⁡wm→∇⁡w almost everywhere in ΩT as m→∞. Therefore, we have wm→w and ∇⁡wm→∇⁡w almost everywhere in ΩT as m→∞. Let v∈Lp⁢(0,T;V) be arbitrary. Using the continuity of B~k and the fact that (wm,∇⁡wm)→(w,∇⁡w) almost everywhere in ΩT, we obtain

B~k⁢(wm,∇⁡wm)→B~k⁢(w,∇⁡w)

almost everywhere in ΩT as m→∞. We take an arbitrary v∈Lp⁢(0,T;V) and δ>0. Because of the absolute continuity of the Lebesgue integral, we know that there exists an η>0 such that for every set E⊂ΩT with |E|<η we have

∥∇⁡v∥Lp⁢(E)<δ.

Since |ΩT|<∞, we can apply Egoroff’s theorem. For the η chosen above, there exists a set A⊂ΩT with |A|<η such that

limm→∞⁡supΩT∖A⁡|B~k⁢(wm,∇⁡wm)-B~k⁢(w,∇⁡w)|=0.

Therefore, we observe, by applying Hölder’s inequality, the boundedness (4.1) of B~k and |∇⁡wm| in Lp⁢(ΩT), that

|∫0T〈ℬ⁢(wm)-ℬ⁢(w),v〉⁢𝑑t|≤∬ΩT∖A|B~k⁢(wm,∇⁡wm)-B~k⁢(w,∇⁡w)|⁢|∇⁡v|⁢𝑑x⁢𝑑t

+∬A|B~k⁢(wm,∇⁡wm)-B~k⁢(w,∇⁡w)|⁢|∇⁡v|⁢𝑑x⁢𝑑t

≤(∬ΩT∖A|B~k⁢(wm,∇⁡wm)-B~k⁢(w,∇⁡w)|p′⁢𝑑x⁢𝑑t)1p′⁢∥∇⁡v∥Lp⁢(ΩT∖A)

+(∬A|B~k⁢(wm,∇⁡wm)-B~k⁢(w,∇⁡w)|p′⁢𝑑x⁢𝑑t)1p′⁢∥∇⁡v∥Lp⁢(A)

≤|Ω|1p′⁢supΩT∖A⁡|B~k⁢(wm,∇⁡wm)-B~k⁢(w,∇⁡w)|⁢∥∇⁡v∥Lp⁢(ΩT)

+∥B~k⁢(wm,∇⁡wm)-B~k⁢(w,∇⁡w)∥Lp′⁢(ΩT)⁢∥∇⁡v∥Lp⁢(A)

≤|ΩT|1p′⁢∥∇⁡v∥Lp⁢(ΩT)⁢supΩT∖A⁡|B~k⁢(wm,∇⁡wm)-B~k⁢(w,∇⁡w)|+C⁢δ→C⁢δ

as m→∞. Since δ>0 was arbitrary, this proves that ℬ⁢(wm)⇀ℬ⁢(w) in Lp′⁢(0,T;V′) as m→∞. Since weak limits are well defined, we have ϱ=ℬ⁢(w), and the lemma is proved. ∎

Next we show that the weak formulation (4.4) also holds for Steklov means.

Lemma 4.4.

The weak formulation (4.4) also holds for the Steklov means given in Definition 3.3, i.e.

(4.12)∫ab∫Ω[[Bk(vk,∇vk)]h⋅∇φ+∂t[vk]hφ]dxdt=∫ab∫Ω[[f]hφ+Bk(0,0)⋅∇φ]dxdt

for all 0≤a<b≤T-h and all

φ∈Lp⁢(0,T;W01,p⁢(Ω))∩L∞⁢(ΩT).

Proof.

We can obtain this by applying Fubini’s theorem and testing (4.4) with the function [φ⁢(x,t)⁢ξ⁢(t)]h¯, where φ∈C∞⁢(Ω¯×[0,T]) vanishes outside of compact subsets of Ω and ξ∈C0∞⁢(0,T-h). A similar equation holds for the reversed Steklov mean. ∎

For the next lemma, we use a comparison principle argument as in [3].

Lemma 4.5.

Suppose k>1 and vk is a weak solution to the Cauchy–Dirichlet problem (4.3) in the sense of Definition 4.2. Then the solution vk is bounded from below, i.e. vk≥1k almost everywhere in ΩT.

Proof.

We use the test function φ=ζε⁢(1k-vk), where

ζε⁢(s):={0,s<0,sε,s∈[0,ε],1,s>ε,

in the weak form with respect to the reversed Steklov means as in Lemma 4.4. Since ζε and ζε′ are bounded, the function φ is indeed an admissible test function. We define the primitive of ζε:

Zε⁢(s):=∫0sζε⁢(σ)⁢𝑑σ={0,s<0,s22⁢ε,s∈[0,ε],s-ε2,s>ε.

The convexity of Zε together with the properties for the Steklov mean gives

∂t[Zε(1k-vk)]h¯(x,t)=1h[Zε(1k-vk)(x,t)-Zε(1k-vk)(x,t-h)]

≤1h⁢ζε⁢(1k-vk)⁢(x,t)⁢[(1k-vk)⁢(x,t)-(1k-vk)⁢(x,t-h)]

=-1h⁢ζε⁢(1k-vk)⁢(x,t)⁢[vk⁢(x,t)-vk⁢(x,t-h)]

=-φ(x,t)∂t[vk]h¯(x,t).

We use the fact that f≥0 together with the estimate above and take the limit h↓0 to obtain that

(4.13)[∫ΩZε⁢(1k-vk)⁢𝑑x]ab+∫ab∫Ω[Bk⁢(0,0)-Bk⁢(vk,∇⁡vk)]⋅∇⁡φ⁢d⁢x⁢d⁢t≤0

for almost every 0<a<b<T. We calculate

∇⁡φ=-ε-1⁢χ{1k-ε<vk<1k}⁢∇⁡vk.

Therefore, only the case vk<1k is relevant, since in the other case ∇⁡φ=0. We apply the monotonicity of B and observe

-[Bk⁢(0,0)-Bk⁢(vk,∇⁡vk)]⋅∇⁡vk=[Bk⁢(vk,∇⁡vk)-Bk⁢(0,0)]⋅∇⁡vk

≥ν⁢k-α⁢(|β⁢k1-β⁢∇⁡vk+∇⁡z|2+|∇⁡z|2)p-22⁢|∇⁡vk|2

≥0.

So the second integral on the left-hand side of (4.13) is nonnegative and it can be estimated by zero from below. We have

∫ΩZε⁢(1k-vk)⁢(x,b)⁢𝑑x≤∫ΩZε⁢(1k-vk)⁢(x,a)⁢𝑑x,

for almost every 0<a<b<T. As we have proved in Lemma 4.3,

vk∈C⁢([0,T];L2⁢(Ω)) and vk⁢(0)=1k+𝚿.

Therefore, by taking a↓0 and ε↓0, we find that

0≤∫Ω(1k-vk)+⁢(x,b)⁢𝑑x=0

for almost every 0<b<T. This gives 1k≤vk almost everywhere in ΩT. ∎

Remember the truncation Tk:ℝ→ℝ defined by Tk⁢(s):=min⁡{k,max⁡{s,1k}}. We want to show that the approximative solutions vk are bounded in the L∞-norm by a constant independent of k. Therefore, we define the function

(4.14)v~k:=Tk∘vk.

For s≥1k, we observe that Tk⁢(s)=min⁡{s,k}. Thus, by Lemma 4.5, v~k=min⁡{vk,k}. The proof of the next lemma is similar to [3, Lemma 5.4]. For the sake of completeness, we include it.

Lemma 4.6.

We assume that k>1 and vk is an admissible weak solution to the Cauchy–Dirichlet problem (4.3) in the sense of Definition 4.2. Let further be M≥supΩ¯⁡Ψ+1. Then the function v~k, defined in (4.14), satisfies

supτ∈[0,T]∫Ω(v~kβ+12-Mβ+12)+2(x,τ)dx+∬ΩT|∇(v~kβ-Mβ)+|pdxdt

(4.15)≤c⁢∬ΩT∩{v~k>M}[|∇⁡z|β⁢p+fp′+Mβ⁢p]⁢𝑑x⁢𝑑t

for a constant c=c⁢(β,p,Ω) which does not depend on k.

Proof.

We distinguish two cases. In the case k≤M, the inequality holds since v~k≤k≤M. For the case k>M, we define the function

ξτ,δ⁢(t):={0,t<0,1δ⁢t,t∈[0,δ],1,t∈[δ,τ],1-1δ⁢(t-τ),t∈[τ,τ+δ],0,t>τ+δ,

for 0<δ<τ<τ+δ<T. We use the formulation with the Steklov means (4.12) with a=0, b=T-h and φ=ξτ,δ⁢(Tk⁢([vk]h)β-Mβ). We take h so small that the factor ξτ,δ is supported in [0,T-h]. Then φ is an admissible test function since

φ∈Lp⁢(0,T;W01,p⁢(Ω))∩L∞⁢(ΩT).

Now we define

ζ⁢(s):=(Tk⁢(s)β-Mβ)+

and

Z⁢(s):=∫0sζ⁢(t)⁢𝑑t={0,s≤M,𝔟⁢[s,M],s∈[M,k],𝔟⁢[k,M]+(kβ-Mβ)⁢(s-k),s>k,

and observe that φ=ξτ,δ⁢ζ⁢([vk]h). We have defined the boundary term 𝔟 in Notation 3.1. We take the limits h↓0 and δ↓0 in (4.12) and obtain that

∫ΩZ(vk)(x,τ)dx+∬ΩτBk(vk,∇vk)⋅∇(v~kβ-Mβ)+dxdt

(4.16)=∬Ωτ[f(v~kβ-Mβ)++Bk(0,0)⋅∇(v~kβ-Mβ)+]dxdt

for every τ∈(0,T]. At first, we consider the set

{(x,t)∈ΩT:vk⁢(x,t)≤k}.

Note that on

{(x,t)∈ΩT:vk⁢(x,t)≤M}

we have ∇(v~kβ-Mβ)+=0 almost everywhere. We observe, by applying the monotonicity (4.2), the boundedness (4.1) and Young’s inequality, that

Bk(vk,∇vk)⋅∇(v~kβ-Mβ)+=βv~kβ-1Bk(vk,∇vk)⋅∇v~kχ{v~k>M}

≥ν⁢β1-p⁢χ{v~k>M}⁢|∇⁡v~kβ+β⁢v~kβ-1⁢∇⁡z|p

-C⁢β2-p⁢v~kβ-1+α⁢χ{v~k>M}⁢(v~k(1-β)⁢(p-1)⁢|∇⁡v~kβ+β⁢v~kβ-1⁢∇⁡z|p-1⁢|∇⁡z|)

≥c1⁢χ{v~k>M}⁢|∇⁡v~kβ|p-c2⁢χ{v~k>M}⁢v~kp⁢(β-1)⁢|∇⁡z|p

=c1|∇(v~kβ-Mβ)+|p-c2χ{v~k>M}v~kp⁢(β-1)|∇z|p,

where c1:=c⁢(β,p,ν) and c2:=c⁢(β,p) both do not depend on k. On the other hand, we observe, by applying (3.1), that

Z⁢(vk)≥χ{v~k>M}⁢𝔟⁢[v~k,M]≥c⁢(β)⁢(v~kβ+12-Mβ+12)+2.

We estimate the second term on the right-hand side of (4.16) using the Cauchy–Schwarz inequality, Young’s inequality for ε>0 and the fact that k>1:

Bk(0,0)⋅∇(v~kβ-Mβ)+≤|Bk(0,0)||∇(v~kβ-Mβ)+|

≤ε|∇(v~kβ-Mβ)+|p+cχ{v~k>M}|Bk(0,0)|pp-1

≤ε|∇(v~kβ-Mβ)+|p+cχ{v~k>M}v~kp⁢(β-1)|∇z|p.

Thus, by including the integral over the first term on the right-hand side in the second integral on the left-hand side of (4.16), we find that

∫Ω(v~kβ+12-Mβ+12)+2(x,τ)dx+∬Ωτ|∇(v~kβ-Mβ)+|pdxdt≤c∬Ωτ∩{v~k>M}[v~kp⁢(β-1)|∇z|p+f(v~kβ-Mβ)]dxdt

for any τ∈(0,T] and a constant c>0 not depending on k. Because

(v~kβ-Mβ)+∈Lp⁢(0,T;W01,p⁢(Ω)),

we can apply Young’s and Poincaré’s inequality and see that

∫Ω(v~kβ+12-Mβ+12)+2(x,τ)dx+∬Ωτ|∇(v~kβ-Mβ)+|pdxdt≤c∬Ωτ∩{v~k>M}[|∇z|β⁢p+fp′+Mβ⁢p]dxdt.

Taking the supremum over the first term on the left-hand side and choosing τ=T for the other terms, we obtain (4.15). ∎

The proof of the next corollary can be achieved completely analogously to [3, Corollary 5.5].

Corollary 4.7.

We assume that k>1 and let vk be an admissible weak solution to the Cauchy–Dirichlet problem (4.3) in the sense of Definition 4.2. Then there is a constant c>0 depending only on β, p and the domain Ω such that the function v~k defined in (4.14) satisfies

∬ΩTv~kβ⁢p⁢𝑑x⁢𝑑t≤c⁢K,

where

K:=∬ΩT[|∇⁡z|β⁢p+fp′]⁢𝑑x⁢𝑑t+(supΩ¯⁡𝚿β⁢p+1)⁢|ΩT|.

The next lemma gives us the boundedness of vk in the L∞-norm. The proof uses a De Giorgi type iteration as in [3, Lemma 5.6].

Lemma 4.8.

Suppose k>1 and let vk be an admissible weak solution to the Cauchy–Dirichlet problem (4.3) in the sense of Definition 4.2. Then there is a constant L>0 depending only on n, β, p, ΩT, f, Ψ, z, and σ (and thus independent of k) such that for every k>L we have

vk≤L almost everywhere in ⁢ΩT.

With Lemma 4.5, Lemma 4.8 and the definition of Bk, we find that

Bk⁢(vk,∇⁡vk)=B⁢(vk,∇⁡vkβ) almost everywhere in ⁢ΩT

if k is large enough. Since for every k>L we have 1k≤vk≤L<k almost everywhere in ΩT, thus

Bk⁢(vk,∇⁡vk)=B⁢(vk,β⁢vkβ-1⁢∇⁡vk)=B⁢(vk,∇⁡vkβ)

almost everywhere in ΩT. We obtain, by applying the weak form of the differential equation (4.4), that vk solves the integral equation

(4.17)∬ΩT[B⁢(vk,∇⁡vkβ)⋅∇⁡φ-vk⁢∂t⁡φ]⁢𝑑x⁢𝑑t=∬ΩT[f⁢φ+Bk⁢(0,0)⋅∇⁡φ]⁢𝑑x⁢𝑑t

for any φ∈C0∞⁢(ΩT). The next lemma gives us a version of (4.17) for test functions which are not necessarily zero for time zero.

Lemma 4.9.

For large enough k and φ∈C∞⁢(Ω¯×[0,T]) with support in K×[0,τ], where K⊂Ω is compact and τ∈(0,T), the following integral equation holds true:

∬ΩT[B⁢(vk,∇⁡vkβ)⋅∇⁡φ-vk⁢∂t⁡φ]⁢𝑑x⁢𝑑t=∬ΩT[f⁢φ+Bk⁢(0,0)⋅∇⁡φ]⁢𝑑x⁢𝑑t+∫Ωvk⁢(0)⁢φ⁢(0)⁢𝑑x.

Proof.

We define the function

ξδ⁢(t):={1δ⁢t,t∈[0,δ],1,t>δ.

We take an arbitrary φ∈C∞⁢(Ω¯×[0,T]) with compact support in K×[0,τ], where K⊂Ω is compact and τ∈(0,T). Observing that ξδ⁢φ∈Lp⁢(0,T;W01,p⁢(Ω)), we use ξδ⁢φ as a test function in (4.17). We let δ↓0 and apply Lebesgue’s differentiation theorem to obtain the desired result. ∎

The next lemma can be proved similarly to Lemma 3.7.

Lemma 4.10.

Suppose k>1 and let vk be an admissible weak solution to the Cauchy–Dirichlet problem (4.3) in the sense of Definition 4.2. For all

wβ∈k-β+Lp⁢(0,T;W01,p⁢(Ω)) with ⁢∂t⁡wβ∈Lβ+1β⁢(ΩT),

and any ζ∈W1,∞⁢([0,T];R≥0) satisfying ζ⁢(0)=0=ζ⁢(T), we obtain the following modified weak form of the partial differential equation:

∬ΩTζ′⁢𝔟⁢[vk,w]⁢𝑑x⁢𝑑t=∬ΩTζ⁢[∂t⁡wβ⁢(vk-w)+B⁢(vk,∇⁡vkβ)⋅(∇⁡vkβ-∇⁡wβ)]⁢𝑑x⁢𝑑t

(4.18)-∬ΩTζ⁢[f⁢(vkβ-wβ)+Bk⁢(0,0)⋅(∇⁡vkβ-∇⁡wβ)]⁢𝑑x⁢𝑑t.

The last lemma of this section shows that the gradient of vkβ is bounded in Lp. In the proof, we proceed similarly to [3, Lemma 5.9].

Lemma 4.11.

Suppose that k>1 and let vk be an admissible weak solution to the Cauchy–Dirichlet problem (4.3) in the sense of Definition 4.2. Then there is a constant C>0 such that for all k∈N we obtain

∬ΩT|∇⁡vkβ|p⁢𝑑x⁢𝑑t≤C.

The constant C depends only on n, β, p, ΩT, f, Ψ, and z.

Proof.

The boundedness and monotonicity in Lemma 4.1, Lemma 4.8 and Young’s inequality imply that

Bk⁢(vk,∇⁡vkβ)⋅∇⁡vkβ≥c⁢|∇⁡vkβ|p-c~⁢vkp⁢(β-1)⁢|∇⁡z|p≥c⁢|∇⁡vkβ|p-c~⁢Lβ-1⁢|∇⁡z|p,

where the constants c>0 and c~>0 do not depend on k. We take an arbitrary δ∈(0,T2). Then we set

ξδ⁢(t):={1δ⁢t,t∈[0,δ],1,t∈[δ,T-δ],1δ⁢(T-t),t∈[T-δ,T].

We use ζ=ξδ and w=1k in (4.18). Because ∂t⁡w=0, the first integral on the right-hand side vanishes. The boundedness in Lemma 3.1, the dominated convergence theorem and Lemma 4.8 together with the estimate from above imply

∬ΩT|∇⁡vkβ|p⁢𝑑x⁢𝑑t≤cT⁢∫Ω|∇⁡z|p⁢𝑑x+c⁢∫Ω(𝚿+1)β+1⁢𝑑x+c⁢Lβ⁢∬ΩTf⁢𝑑x⁢𝑑t+c⁢∬ΩT|∇⁡z|p-1⁢|∇⁡vkβ|⁢𝑑x⁢𝑑t.

We apply Young’s inequality to the last integral on the right-hand side and observe

∬ΩT|∇⁡vkβ|p⁢𝑑x⁢𝑑t≤c⁢[∫Ω[|∇⁡z|p+(𝚿+1)β+1]⁢𝑑x+Lβ⁢∬ΩTf⁢𝑑x⁢𝑑t].

This proves the claim. ∎

5 The main existence proof

Since Lp⁢(0,T;W1,p⁢(Ω)) is a reflexive Banach space and (vkβ) is a bounded sequence in Lp⁢(0,T;W1,p⁢(Ω)) because of Lemma 4.8 and Lemma 4.11, there exists a subsequence (vkjβ) of (vkβ) converging weakly in Lp⁢(0,T;W1,p⁢(Ω)). The limit-function vβ is nonnegative because every vkβ is nonnegative. The next lemma gives us strong convergence of a subsequence (vkj) of (vk).

Lemma 5.1.

Suppose (vk)k∈N are weak solutions to the Cauchy–Dirichlet problems (4.3) in the sense of Definition 4.2. Then there is a subsequence (vkj) of (vk) and a nonnegative function v∈L∞⁢(ΩT) with

vβ∈Lp⁢(0,T;W01,p⁢(Ω))

such that

{vkjβ⇀vβweakly in ⁢Lp⁢(0,T;W1,p⁢(Ω)),vkj→vstrongly in ⁢Lq⁢(ΩT)⁢ for any ⁢q≥1⁢ and almost everywhere in ⁢ΩT.

Proof.

Because of Lemma 4.8 and Lemma 4.11, (vkβ-k-β) is a bounded sequence in Lp⁢(0,T;W01,p⁢(Ω)). Thus, we know that there exists a subsequence (vkj) of (vk) and a limit-function w∈Lp⁢(0,T;W01,p⁢(Ω)) such that vkjβ-kj-β⇀w weakly in Lp⁢(0,T;W1,p⁢(Ω)). This also implies vkjβ⇀w weakly in Lp⁢(0,T;W1,p⁢(Ω)). We show that (vkj) converges strongly. For this we define the function

ξδ⁢(t):={0,t<τ-δ,1δ⁢(t-τ+δ),t∈[τ-δ,τ],1,t∈(τ,τ+δ),1δ⁢(τ+h+δ-t),t∈[τ+h,τ+h+δ],0,t>τ+h+δ,

for τ∈(0,T), h∈(0,T-h) and δ∈(0,min⁡{τ,T-τ-h}). Now we take φ∈C0∞⁢(Ω) and use ξδ⁢φ as a test function in (4.17). We apply the boundedness of B, Lemma 4.11, Lemma 4.8 and Young’s inequality to see that

∫Ω[vk⁢(τ)-vk⁢(τ+h)]⁢φ⁢𝑑x=∫ττ+h∫Ω[B⁢(vk,∇⁡vkβ)⋅∇⁡φ-f⁢φ-Bk⁢(0,0)⋅∇⁡φ]⁢𝑑x⁢𝑑t.

Interpreting vk⁢(t) as an element of (W01,p⁢(Ω))′ for t∈(0,T), letting 〈⋅,⋅〉 denote the dual pairing of (W01,p⁢(Ω))′ and W01,p⁢(Ω) and using the boundedness of Bk⁢(0,0), we thus have

|〈vk⁢(τ)-vk⁢(τ-h),φ〉|≤∫ττ+h∫Ω[(|B⁢(vk,∇⁡vkβ)|+|Bk⁢(0,0)|)⁢|∇⁡φ|+|f|⁢|φ|]⁢𝑑x⁢𝑑t

≤c⁢∥φ∥W1,p⁢(Ω)⁢h1p⁢[∬ΩT[|∇⁡vkβ|p+|∇⁡z|p+fp-1p]⁢𝑑x⁢𝑑t]p-1p

(5.1)≤C⁢h1p⁢∥φ∥W1,p⁢(Ω)

for all φ∈C0∞⁢(Ω), where c=c⁢(α,p,L) and we have applied Lemma 4.11 in the last line. Since C0∞⁢(Ω) is dense in W01,p⁢(Ω), inequality (5.1) also holds for all φ∈W01,p⁢(Ω). Therefore, we can estimate the dual norm as follows:

∥vk⁢(τ)-vk⁢(τ+h)∥W-1,p′⁢(Ω)=supφ∈W01,p⁢(Ω)φ≠0⁡C⁢h1p⁢∥φ∥W1,p⁢(Ω)∥φ∥W1,p⁢(Ω)=C⁢h1p.

As we have seen before, (vk)k∈ℕ is bounded in L∞⁢(ΩT) and (vkβ)k∈ℕ is bounded in Lp⁢(0,T;W1,p⁢(Ω)). We apply [3, Corollary 4.6 (i)] for Y=W-1,p′⁢(Ω), m=β, θ=β⁢p, q=μ=p, and F=(vkj)j∈ℕ. Thus, (vkjβ) is relatively compact in Lp⁢(ΩT). Therefore, any subsequence of (vkjβ) has a convergent subsequence. These convergent subsequences all converge to w, because vkjβ⇀w and weak and strong limits coincide. Thus, the whole sequence (vkjβ) converges strongly to w in Lp⁢(ΩT). If we pass to another subsequence (again denoted by (vkj)), we see that vkjβ→w almost everywhere in ΩT. Since the vk are bounded almost everywhere, we observe that w∈L∞⁢(ΩT). Now we take v∈L∞⁢(ΩT) such that vβ=w. This tells us that vkjβ⇀vβ weakly in Lp⁢(0,T;W1,p⁢(Ω)) as j→∞. Mazur’s theorem gives us vβ∈Lp⁢(0,T;W01,p⁢(Ω)). Further, vkjβ→vβ strongly in Lp⁢(ΩT) as j→∞. By case distinction, strong convergence in Lq⁢(ΩT), for q≥1, can be obtained. ∎

The next lemma also gives us a strong convergence for the gradients.

Lemma 5.2.

Suppose that (vk) are weak solutions to the Cauchy–Dirichlet problem (4.3) in the sense of Definition 4.2. Then the subsequence (vkj) of (vk) satisfies

∇⁡vkjβ→∇⁡vβ strongly in ⁢Lp⁢(Ω×J),

for any closed subinterval J of (0,T).

Proof.

Let ξ∈C0∞⁢((0,T);[0,1]). The monotonicity in Lemma 3.1, Lemma 4.8 and Lemma 4.11 imply similarly to the proof of Lemma 4.3 that

1c⁢[∬ΩTξ⁢|∇⁡vkβ-∇⁡vβ|p⁢𝑑x⁢𝑑t]γ≤∬ΩTξ⁢B⁢(vk,∇⁡vkβ)⋅(∇⁡vkβ-∇⁡vβ)⁢𝑑x⁢𝑑t-∬ΩTξ⁢B⁢(v,∇⁡vβ)⋅(∇⁡vkβ-∇⁡vβ)⁢𝑑x⁢𝑑t

+∬ΩTξ⁢[B⁢(v,∇⁡vβ)-B⁢(vk,∇⁡vβ)]⋅(∇⁡vkβ-∇⁡vβ)⁢𝑑x⁢𝑑t

(5.2)=:Ik+IIk+IIIk,

where γ=max⁡{1,2p} and the constant c>0 is independent of k. It is easy to see that ξ⁢B⁢(v,∇⁡vβ)∈Lp′⁢(ΩT;ℝn). Then the weak convergence vkjβ⇀vβ in Lp⁢(0,T;W1,p⁢(Ω)), proved in Lemma 5.1, shows that the second term on the right-hand side of (5.2) converges to zero as j→∞. The boundedness of B, Lemma 4.8, the fact that vkj→v almost everywhere in ΩT and Hölder’s inequality guarantee that

limj→∞⁡I⁢I⁢Ikj≡limj→∞⁡∬ΩTξ⁢[B⁢(v,∇⁡vβ)-B⁢(vkj,∇⁡vβ)]⋅(∇⁡vkjβ-∇⁡vβ)⁢𝑑x⁢𝑑t=0.

Now we calculate the limit of Ik. We rewrite Ik as

Ik=∬ΩTξB(vk,∇vkβ)⋅(∇vkβ-∇〚vβ〛h)dxdt+∬ΩTξB(vk,∇vkβ)⋅(∇〚vβ〛h-∇vβ)dxdt

=:Ik(1)+Ik(2),

where 〚vβ〛h is as in Lemma 3.3. Using the fact that B⁢(vk,∇⁡vkβ) is bounded in Lp′ and Hölder’s inequality, we observe that

|Ik(2)|≤c∥∇〚vβ〛h-∇vβ∥Lp⁢(ΩT)

for a constant c>0 independent of k. To show that the term Ik(1) converges, we use the modified weak formulation (4.18) with the comparison function wh,kβ=k-β+〚vβ〛h. We obtain

Ik(1)=∬ΩT[ξ′𝔟[vk,wh,k]-ξ∂t〚vβ〛h(vk-wh,k)]dxdt

(5.3)+∬ΩTξ[f(vkβ-wh,kβ)+Bk(0,0)⋅(∇vkβ-∇〚vβ〛h)]dxdt.

We apply the boundedness of Bk⁢(0,0), Lemma 4.11 and the fact that the exponential mollification preserves the space:

∬ΩTξ[Bkj(0,0)⋅(∇vkjβ-∇〚vβ〛h)]dxdt≤ckj(1-β)⁢(p-1)∬ΩT|∇z|p-1(|∇vkjβ|+|∇〚vβ〛h|)dxdt

≤c⁢kj(1-β)⁢(p-1)

→0

as j→∞. Using the fact that Ikj+I⁢I⁢Ikj→0 as j→∞ and the convergence of the remaining terms in (5.3), we have

lim supj→∞[∬ΩTξ|∇vkjβ-∇vβ|pdxdt]γ≤c∥∇〚vβ〛h-∇vβ∥Lp⁢(ΩT)+c∬ΩT[ξ′𝔟[v,(〚vβ〛h)1β]+ξf(vβ-〚vβ〛h)]dxdt.

Because of Lemma 3.4 (iii) and Hölder’s inequality, both terms on the right-hand side converge to zero as h↓0. In total, this implies

limj→∞⁡∬ΩTξ⁢|∇⁡vkjβ-∇⁡vβ|p⁢𝑑x⁢𝑑t=0.

For any closed subinterval J⊂(0,T), we can choose ξ∈C0∞⁢(0,T;[0,1]) such that ξ|J=1. Then χJ≤ξ, and we obtain the desired result. ∎

We show that the function v given in Lemma 5.1 satisfies the following weak formulation for test functions which do not have to vanish in zero.

Lemma 5.3.

Let v be as in Lemma 5.1. Then the following integral equation holds true:

(5.4)∬ΩT[B⁢(v,∇⁡vβ)⋅∇⁡φ-v⁢∂t⁡φ]⁢𝑑x⁢𝑑t=∬ΩTf⁢φ⁢𝑑x⁢𝑑t+∫Ω𝚿⁢φ⁢(0)⁢𝑑x

for every φ∈C∞⁢(Ω¯×[0,T]) with support contained in K×[0,τ], where K⊂Ω is compact and τ∈(0,T).

Proof.

We take an arbitrary test function φ∈C∞(Ω¯×[0,T])) with support in K×[0,τ], where K⊂Ω is compact, τ∈(0,T) and |∇⁡φ|≤1. Because of Lemma 5.1 and Lemma 5.2, we can extract another subsequence, which we also denote by (vkj), such that vkj→v and ∇⁡vkj→∇⁡v almost everywhere in Ω×I as j→∞, where I is an arbitrary closed subinterval of (0,T). We want to take the limit j→∞. Because of Lemma 5.1, we have vkj→v in L1⁢(ΩT) as j→∞, and thus

|∬ΩT(vkj-v)⁢∂t⁡φ⁢d⁢x⁢d⁢t|≤∬ΩT|vkj-v|⁢|∂t⁡φ|⁢𝑑x⁢𝑑t

≤supΩ¯×[0,T]⁡|∂t⁡φ|⁢∬ΩT|vkj-v|⁢𝑑x⁢𝑑t

→0

as j→0. Further, we obtain, by applying the boundedness of Bk⁢(0,0), that

|∬ΩTBk⁢(0,0)⋅∇⁡φ⁢d⁢x⁢d⁢t|≤∬ΩT|Bk⁢(0,0)|⁢|∇⁡φ|⏟≤1⁢𝑑x⁢𝑑t

≤c⁢kj(1-β)⁢(p-1)⁢T⁢∫Ω|∇⁡z|p-1⁢𝑑x

→0

as j→∞. At last, we obtain that

limj→∞⁡∫Ωvkj⁢(0)⁢φ⁢(0)⁢𝑑x=limj→∞⁡∫Ω(1kj+𝚿)⁢φ⁢(0)⁢𝑑x

=limj→∞⁡1kj⁢∫Ωφ⁢(0)⁢𝑑x+∫Ω𝚿⁢φ⁢(0)⁢𝑑x

=∫Ω𝚿⁢φ⁢(0)⁢𝑑x.

We define the difference of the vector fields

ℬk:=B⁢(vk,∇⁡vkβ)-B⁢(v,∇⁡vβ).

For an arbitrary δ∈(0,τ), we find that

∬ΩT|B⁢(vk,∇⁡vkβ)⋅∇⁡φ-B⁢(v,∇⁡vβ)⋅∇⁡φ|⁢𝑑x⁢𝑑t≤∬Ω×[0,δ]|ℬk|⁢𝑑x⁢𝑑t+∬Ω×[δ,τ]|ℬk|⁢𝑑x⁢𝑑t.

For arbitrary ε>0, we choose δ>0 such that

∬Ω×[0,δ]|ℬk|⁢𝑑x⁢𝑑t≤ε2.

With such a fixed δ, we now estimate the second term on the right-hand side. As stated in the beginning, we know that (vkj,∇⁡vkjβ)→(v,∇⁡vβ) almost everywhere in Ω×[δ,τ] as j→∞. This implies, because of the continuity of B, that

|B⁢(vkj,∇⁡vkjβ)-B⁢(v,∇⁡v)|→0

almost everywhere in Ω×[δ,τ]. The boundedness of |ℬk| in Lp′⁢(ΩT) together with Egoroff’s theorem justifies the existence of a set A⊂Ω×[δ,τ] such that

∬Ω×[δ,τ]|ℬkj|⁢𝑑x⁢𝑑t=∬(Ω×[δ,τ])∖A|ℬkj|⁢𝑑x⁢𝑑t+∬A|ℬkj|⁢𝑑x⁢𝑑t

≤|ΩT|⁢sup(Ω×[δ,τ])∖A⁡|B⁢(vkj,∇⁡vkjβ)-B⁢(v,∇⁡vβ)|+|A|1p⁢[∬ΩT|ℬkj|p′⁢𝑑x⁢𝑑t]1p′

≤|ΩT|⁢sup(Ω×[δ,τ])∖A⁡|B⁢(vkj,∇⁡vkjβ)-B⁢(v,∇⁡vβ)|+ε2→j→∞ε2.

In total, since ε>0 was arbitrary, we have

limj→∞⁡∬ΩT|B⁢(vkj,∇⁡vkjβ)⋅∇⁡φ-B⁢(v,∇⁡vβ)⋅∇⁡φ|⁢𝑑x⁢𝑑t=0.

The proof is finished. ∎

The proof of Theorem 2.2 is analogous to [3, Theorem 2.2].

Proof of Theorem 2.2.

We show that the function v obtained in Lemma 5.1 is a weak solution to the Cauchy–Dirichlet problem (2.6) in the sense of Definition 2.1. Since the test functions φ in (2.7) are in C0∞⁢(ΩT), the second integral on the right-hand side of (5.4) vanishes. This shows (2.7). By Lemma 3.8, we know that v∈C⁢([0,T];Lβ+1⁢(Ω)). We use (5.4) with the test function φ⁢ζε where φ∈C0∞⁢(Ω) and

ζε⁢(t):={1ε⁢(φ-t),t∈[0,ε],0,t≥ε.

We pass to the limit ε↓0 and observe 𝚿=v⁢(0). ∎

Communicated by Frank Duzaar

Funding source: Austrian Science Fund

Award Identifier / Grant number: P 31956

Acknowledgements

The author has been supported by the FWF-Project P 31956 “Doubly Nonlinear Evolution Equations”.

References

[1] R. Alonso, M. Santillana and C. Dawson, On the diffusive wave approximation of the shallow water equations, European J. Appl. Math. 19 (2008), no. 5, 575–606. 10.1017/S0956792508007675Suche in Google Scholar

[2] H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z. 183 (1983), no. 3, 311–341. 10.1007/BF01176474Suche in Google Scholar

[3] V. Bögelein, N. Dietrich and M. Vestberg, Existence of solutions to a diffusive shallow medium equation, J. Evol. Equ. 21 (2021), no. 1, 845–889. 10.1007/s00028-020-00604-ySuche in Google Scholar PubMed PubMed Central

[4] V. Bögelein, F. Duzaar, R. Korte and C. Scheven, The higher integrability of weak solutions of porous medium systems, Adv. Nonlinear Anal. 8 (2019), no. 1, 1004–1034. 10.1515/anona-2017-0270Suche in Google Scholar

[5] V. Bögelein, F. Duzaar and P. Marcellini, Existence of evolutionary variational solutions via the calculus of variations, J. Differential Equations 256 (2014), no. 12, 3912–3942. 10.1016/j.jde.2014.03.005Suche in Google Scholar

[6] V. Bögelein, F. Duzaar, P. Marcellini and C. Scheven, Doubly nonlinear equations of porous medium type, Arch. Ration. Mech. Anal. 229 (2018), no. 2, 503–545. 10.1007/s00205-018-1221-9Suche in Google Scholar

[7] Y. Cai and S. Zhou, Existence and uniqueness of weak solutions for a non-uniformly parabolic equation, J. Funct. Anal. 257 (2009), no. 10, 3021–3042. 10.1016/j.jfa.2009.08.007Suche in Google Scholar

[8] J. I. Diaz and F. de Thélin, On a nonlinear parabolic problem arising in some models related to turbulent flows, SIAM J. Math. Anal. 25 (1994), no. 4, 1085–1111. 10.1137/S0036141091217731Suche in Google Scholar

[9] E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer, New York, 1993. 10.1007/978-1-4612-0895-2Suche in Google Scholar

[10] A. V. Ivanov, P. Z. Mkrtychyan and V. Yaeger, Existence and uniqueness of a regular solution of the Cauchy–Dirichlet problem for a class of doubly nonlinear parabolic equations, J. Math. Sci. 84 (1997), 845–855. 10.1007/BF02399936Suche in Google Scholar

[11] J. Kinnunen and P. Lindqvist, Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation, Ann. Mat. Pura Appl. (4) 185 (2006), no. 3, 411–435. 10.1007/s10231-005-0160-xSuche in Google Scholar

[12] A. Mielke and U. Stefanelli, Weighted energy-dissipation functionals for gradient flows, ESAIM Control Optim. Calc. Var. 17 (2011), no. 1, 52–85. 10.1051/cocv/2009043Suche in Google Scholar

[13] M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations 103 (1993), no. 1, 146–178. 10.1006/jdeq.1993.1045Suche in Google Scholar

[14] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Math. Surveys Monogr. 49, American Mathematical Society, Providence, 1997. Suche in Google Scholar

[15] T. Singer and M. Vestberg, Local boundedness of weak solutions to the diffusive wave approximation of the shallow water equations, J. Differential Equations 266 (2019), no. 6, 3014–3033. 10.1016/j.jde.2018.08.051Suche in Google Scholar

[16] T. Singer and M. Vestberg, Local Hölder continuity of weak solutions to a diffusive shallow medium equation, Nonlinear Anal. 185 (2019), 306–335. 10.1016/j.na.2019.03.013Suche in Google Scholar

[17] S. Sturm, Existence of weak solutions of doubly nonlinear parabolic equations, J. Math. Anal. Appl. 455 (2017), no. 1, 842–863. 10.1016/j.jmaa.2017.06.024Suche in Google Scholar

[18] V. Vespri, On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations, Manuscripta Math. 75 (1992), no. 1, 65–80. 10.1007/BF02567072Suche in Google Scholar

Received: 2022-12-16

Published Online: 2022-04-20

Published in Print: 2022-07-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

https://doi.org/10.1515/forum-2021-0320

Schlagwörter für diesen Artikel

Shallow medium equation; doubly nonlinear equations; existence; coefficients

Creative Commons

BY 4.0