On Weak Solutions to Parabolic Problem Involving the Fractional p-Laplacian via Young Measures

Ihya Talibi; Farah Balaadich; Brahim El Boukari; Jalila El Ghordaf

doi:10.2478/amsil-2024-0021

1. Introduction

Recently, there has been a lot of interest in the systematic study of problems involving non-local operators due to their frequency in practical real-world applications, such as finance, optimization, soft thin films, stratified materials, and phase transitions. We refer the reader to see [32]. The elliptic theory for linear and quasilinear nonlocal operators has seen extensive research over the past few decades, particularly in the works of Caffarelli and collaborators [4, 5, 14]. Additionally, research on nonlocal nonlinear problems has been extensively explored in [30], we also refer to [9, 10, 11, 15, 22, 24, 25, 26, 31] on related existence results for the problems of elliptic and parabolic type involving non-local fractional Laplacian (p-Laplacian) operators.

In this paper, suppose that Ω is a bounded open domain of ℝⁿ and T is a real positive number. We deal with the following initial boundary value problem:

(1.1)

{\begin{array}{l} \frac{\partial u}{\partial t} + {(- Δ)}_{p}^{s} u = f (x, t, u) & in Q_{T} = Ω \times (0, T), \\ u = 0 & in (ℝ^{n} \ Ω) \times (0, T), \\ u (x, 0) = u (x) & in Ω, \end{array}

where 0 < s < 1 and 2 < p are real numbers, u: Ω × (0, T) → ℝ^m, m ∈ {0, 1, 2, . . .} is a vector-valued function and the function f satisfies the following hypothesis:

(H1) f : Ω × (0, T) × ℝ^m → ℝ^m is a Carathéodory function satisfying
$\begin{matrix} | f (x, t, r) | \leq α_{0} (1 + | r |^{q - 1}), \\ F_{t} (x, t, r) \geq α_{1} (- 1 - | r |^{q}), \end{matrix}$
for all (x, t, r) ∈ Ω × (0, T) × ℝ^m, where α₀, α₁ are positive constants, $F (x, t, r) = \int_{0}^{r} f (x, t, l) dl$ and $F_{t} = \frac{d}{dt} F$ .

The fractional p-Laplacian operator ${(- Δ)}_{p}^{s} u$ is defined as follows:

{(- Δ)}_{p}^{s} u (x, t) = P . V \int_{ℝ^{n}} \begin{matrix} \frac{| u (x, t) - u (y, t) |^{p - 2} (u (x, t) - u (y, t))}{| x - y |^{n + p s}} d y, & x \in ℝ^{n}, \end{matrix}

where P.V stands for “in the principal value sense” and is a frequently used abbreviation. For more information on this operator, see [13].

Concerning the fractional Laplacian (p = 2), a famous model for anomalous diffusion is the following equation: $\frac{\partial u}{\partial t} + {(- Δ)}^{s} u = 0$ , which comes asymptotically from basic random walk models (see [33, 34]). Also in [17], de Pablo et al. proposed the nonlinear anomalous diffusion equation $\frac{\partial u}{\partial t} + {(- Δ)}^{s} (u^{m}) = 0$ , the fractional porous medium equation with 0 < s < 1 and m > 0. We also refer to [34] for more details on this type of equation.

On the other hand, in the case p ≠ 2 and f = 0, Vázquez in [35] proved the existence and uniqueness of strong nonnegative solutions for (1.1). If u₀ ∈ L²(Ω), the existence results of energy solution were studied in [29].

When it comes to the problem (1.1), the existence results are treated in several works, for example, the different issues of the existence and the regularity of energy-weak solutions to the problem same to (1.1) were investigated by Giacomoni et al. in [21]. In [1], the authors have studied the problem (1.1) with f depending only on x and t and proved the existence results with suitable regularity if (f, u₀) ∈ L¹ (Ω_T) × L¹(Ω) and has a nonnegative entropy solution if f₀, u₀ are nonnegative. The same author in [2] proved the asymptotic behavior result of entropy solutions when the right-hand side does not depend on time.

The idea of this work, motivated by all of the results above, is to study the existence of weak solutions to the problem (1.1) by using the Galerkin method combined with the theory of Young measures. To the best of our knowledge, the parabolic problem (1.1) has never been studied by the theory of Young measure. We suggest to the readers to consult [6, 7, 19] which treat some elliptic and parabolic systems by such a theory. In [8], the authors proved the existence of weak solutions to the elliptic case of (1.1) employing the Young measures theory and the Galerkin method.

This article is organized into four sections. In Section 2 we give some background information on fractional Sobolev spaces and a review of the Young measures theory. Later, under some assumptions, we obtain the existence of weak solutions using the Galerkin approximation and the Young measures. The final part is devoted to illustrating the feasibility of the hypotheses with an example.

2. Preliminaries and notations

In this section, we first recall some necessary results which will be used in the next section. Let 1 < p < ∞, s ∈ (0, 1), we define $p_{s}^{*}$ the fractional critical exponent by:

p_{s}^{*} = {\begin{array}{l} \infty & if p s \geq n, \\ n p / (n - p s) & if p s < n . \end{array}

Let Ω ⊂ ℝⁿ be an open set, Q_Ω = (ℝⁿ × ℝⁿ) \(𝒞Ω × 𝒞Ω), Q_τ = Ω × (0, τ) for all τ ∈ (0, T ] and 𝒞Ω = ℝⁿ\Ω. It is clear that Ω × Ω is strictly contained in Q_Ω. W is a linear space of Lebesgue measurable functions from ℝⁿ to ℝ^m such that the restriction to Ω of any function u in W belongs to L^p(Ω; ℝ^m) and

\iint_{Q_{Ω}} \frac{| u (x) - u (y) |^{p}}{| x - y |^{n + p s}} dydx < \infty .

The space W is equipped with the norm

{‖ u ‖}_{W} = {‖ u ‖}_{L^{p} (Ω; ℝ^{m})} + {(\iint_{Q_{Ω}} \frac{| u (x) - u (y) |^{p}}{| x - y |^{n + p s}} dydx)}^{1 / p} .

Let us consider the closed linear subspace

W_{0} = {u \in W : u = 0 a . e . in 𝒞 Ω} .

In W₀, we may also use the norm

{‖ u ‖}_{W_{0}} = {(\iint_{Q_{Ω}} \frac{| u (x) - u (y) |^{p}}{| x - y |^{n + p s}} dydx)}^{1 / p} .

It is known that (W₀, ∥ · ∥_W₀) is a uniformly convex reflexive Banach space (see [36]). The following Poincare’s inequality from [12] will be used below: there exists C_r > 0 such that

(2.1)

\begin{matrix} {‖ ϕ ‖}_{L^{r} (Ω, ℝ^{m})} \leq C_{r} {‖ ϕ ‖}_{W_{0}} & for all & ϕ \in W_{0} & and & r \in [1, p_{s}^{*}] \end{matrix} .

In the sequel, let

p < \frac{n}{s}

and C_i, i = 1, 2, . . . be positive constants that vary from line to line, and are independent of the terms involved in any limit process. We note the following functional space L^p(0, T; W₀), which is a separable and reflexive Banach space endowed with the norm

{‖ u ‖}_{L^{p} (0, T; W_{0})} = {(\int_{0}^{T} {‖ u ‖}_{W_{0}}^{p} dt)}^{1 / p} .

Lemma 2.1. ([20])

The space $𝒞_{0}^{\infty} (Ω; ℝ^{m})$ of infinitely differentiable functions with compact support on Ω is dense in W₀.

Lemma 2.2. ([18])

The following embedding W₀ ↪ L^r (Ω; ℝ^m) is compact for all $r \in [1, p_{s}^{*})$ , and continuous for all $r \in [1, p_{s}^{*}]$ .

In the following, 𝒞₀ (ℝ^m) stands for the space of continuous functions on ℝ^m with compact support with regards to the ∥·∥_∞-norm. The space of signed Radon measures with finite mass is noted ℳ (ℝ^m). The corresponding duality is given by

{μ, ρ 〉 = \int_{ℝ^{m}} ρ (λ) d μ (λ) .

Definition 2.3. ([8])

Let {z_j}_j≥1 be a bounded sequence in L^∞ (Ω; ℝ^m). Then there exist a subsequence {z_k} ⊂ {z_j} and a Borel probability measure µ_x on ℝ^m for almost every x ∈ Ω, such that for a.e. ρ ∈ 𝒞 (ℝ^m) we have $ρ (z_{k}) ⇀ * \bar{ρ}$ weakly in L^∞(Ω), where $\bar{ρ} (x) = {μ_{x}, ρ 〉 = \int_{ℝ^{m}} ρ (λ) d μ_{x} (λ)$ for a.e. x ∈ Ω.

Lemma 2.4. ([23])

Let Ω ⊂ ℝⁿ be Lebesgue measurable (not necessarily bounded) and z_j from Ω to ℝ^m, for j ∈ ℕ, be a sequence of Lebesgue measurable functions. Then there exist a subsequence z_k and a family {µ_x}_x∈Ω of non-negative Radon measures on ℝ^m, such that

(i) ∥µ_x∥_ℳ(ℝ^m) := ∫_ℝ^m dµ_x(λ) ≤ 1 for almost every x ∈ Ω.
(ii) $ρ (z_{k}) ⇀ * \bar{ρ}$ weakly in L^∞(Ω) for all 𝒞₀ (ℝ^m), where $\bar{ρ} (x) = 〈 μ_{x}, ρ 〉$ .
(iii) If for all M > 0
(2.2)
$lim_{N \to \infty} sup_{k \in ℕ} | {x \in Ω \cap B_{M} (0) : | z_{k} (x) | \geq N} | = 0,$
then ∥µ_x∥ = 1 for a.e. x ∈ Ω, and for any measurable Ω′ ⊂ Ω we have $ρ (z_{k}) ⇀ \bar{ρ} = 〈 μ_{x}, ρ 〉$ weakly in L¹ (Ω′) for continuous function ρ provided the sequence ρ (z_k) is weakly precompact in L¹ (Ω′).

3. Local existence of weak solutions

In this section, we define a weak solution to the problem (1.1) and prove the main result (Theorem 3.2 below). We start with the following definition:

Definition 3.1

A function u ∈ L^p(0, T; W₀) is called a weak solution of (1.1), if $\frac{\partial u}{\partial t} \in L^{2} (Q_{T}; ℝ^{m})$ and

\begin{array}{l} \int_{Q_{T}} \frac{\partial u}{\partial t} ϕ dxdt \\ + \int_{0}^{T} \iint_{Q_{Ω}} \frac{| u (x, t) - u (y, t) |^{p - 2} (u (x, t) - u (y, t))}{| x - y |^{n + p s}} (ϕ (x, t) - ϕ (y, t)) dxdydt \\ = \int_{Q_{T}} f (x, t, u) ϕ dxdt, \end{array}

holds for all

ϕ \in C^{1} (0, T; C_{0}^{\infty} (Ω))

Theorem 3.2

If u₀ ∈ W₀, $2 < q < \frac{(2 + p) p_{s}^{*} - 2 p}{p_{s}^{*}} < p_{s}^{*}$ and (H1) is satisfied, then there exists a constant T₀ > 0 such that problem (1.1) has at least one weak solution as T < T₀.

Proof

The proof is divided into three assertions.

Assertion 1: Galerkin approximation

Similar to that in [27], we take a sequence ${w_{j}}_{j \geq 1} \subset C_{0}^{\infty} (Ω; ℝ^{m})$ , such that $C_{0}^{\infty} (Ω; ℝ^{m}) \subset {\bar{\cup_{k \geq 1} U_{k}}}^{C_{1} (\bar{Ω})}$ , where {w_j}_j≥1 is an orthonormal basis in L² (Ω; ℝ^m) and U_k = span {w₁, . . . , w_k}.

Lemma 3.3

For the function u₀ ∈ W₀, there exists a subsequence ξ_k ∈ U_k such that ξ_k → u₀ in W₀ as k → ∞.

Proof

Since u₀ ∈ W₀, we can find a sequence {v_k} in $C_{0}^{\infty} (Ω; ℝ^{m})$ such that v_k → u₀ in W₀. Since ${v_{k}} \subset C_{0}^{\infty} (Ω; ℝ^{m}) \subset {\cup_{M \geq 1} \bar{U_{M}}}^{C^{1} (\bar{Ω}; ℝ^{m})}$ , there exists a sequence ${v_{k}^{i}} \subset \cup_{M \geq 1} U_{M}$ such that $v_{k}^{i} \to v_{k}$ in $C^{1} (\bar{Ω}; ℝ^{m})$ as i tends to ∞. For $\frac{1}{2^{k}}$ , there exists i_k ≥ 1 such that ${‖ v_{k}^{i_{k}} - v_{k} ‖}_{C^{1} (\bar{Ω})} - \leq \frac{1}{2^{k}}$ . Therefore

{‖ v_{k}^{i_{k}} - u_{0} ‖}_{W_{0}} \leq C_{1} {‖ v_{k}^{i_{k}} - v_{k} ‖}_{C^{1} (\bar{Ω})} + {‖ v_{k} - u_{0} ‖}_{W_{0}} .

Hence

v_{k}^{i_{k}} \to u_{0}

in W₀ as k tends to ∞. We denote

u_{k} = v_{k}^{i_{k}}

. Since u_k ∈ U_M≥1 U_M, there exists U_{M_k} such that u_k ∈ U_{M_k}, without loss of generality, we assume that U_M₁ ⊂ U_M₂ as M₁ ≤ M₂. We suppose that M₁ > 1 and define ξ_k as follows:

{\begin{array}{l} ξ_{k} (x) = 0, & for k = 1, \dots, M_{1} - 1, \\ ξ_{k} (x) = u_{1}, & for k = M_{1}, \dots, M_{2} - 1, \\ ξ_{k} (x) = u_{2}, & for k = M_{2}, \dots, M_{3} - 1, \\ ⋮ & ⋮ \end{array}

Then {ξ_k} is the desired sequence such that ξ_k → u₀ in W₀ as k → ∞.

We define the function R_k : [0, T) × ℝ^k → ℝ^k where k is fixed:

\begin{array}{l} {(R (t, ς))}_{i} & = \iint_{Q_{Ω}} \frac{{| \sum_{j = 1}^{k} {(ς_{j} (t))}_{j} w_{j} (x) - \sum_{j = 1}^{k} {(ς_{j} (t))}_{j} w_{j} (y) |}^{p - 2}}{x - y |^{n + p s}} \\ \times (\sum_{j = 1}^{k} {(ς_{j} (t))}_{j} w_{j} (x) - \sum_{j = 1}^{k} {(ς_{j} (t))}_{j} w_{j} (y)) (w_{i} (x) - w_{i} (y)) dxdy, \end{array}

for ς ∈ ℝ^k and i = 1, . . . , k. The function R(t, ς) is continuous in t and ς.

Now, we shall construct the approximating solutions for (1.1) as follows:

u_{k} (x, t) = \sum_{j = 1}^{k} {(b_{j} (t))}_{j} w_{j} (x),

where unknown functions (b(t))_j are determined by the following system of ODE:

(3.1)

{\begin{array}{l} b^{'} (t) + R_{k} (t, b (t)) = S_{k} (t, b (t)), & 0 < t < T, \\ b (0) = ψ_{k} (0), \end{array}

where

{(S_{k} (t, b))}_{i} = \int_{Ω} \begin{matrix} f (x, t, \sum_{j = 1}^{k} b_{j} w_{j}) w_{i} dx, & {(ψ_{k} (0))}_{i} = \int_{Ω} ξ_{k} (x) w_{i} dx \end{matrix},

and

\begin{matrix} ξ_{k} (x) \to u_{0} & in W_{0} & as k \end{matrix} \to \infty where ξ_{k} (x) \in U_{k} .

Multiplying (3.1) by b(t), we get

(3.2)

b^{'} b + R_{k} (t, b) b = S_{k} (t, b) b .

According to (H1), the following inequalities hold

(3.3)

\begin{array}{l} S_{k} (t, b) b & \leq α_{0} \int_{Ω} ({| \sum_{j = 1}^{k} b_{j} w_{j} |}^{q} + | \sum_{j = 1}^{k} b_{j} w_{j} |) dx \\ \leq α_{0} \int_{Ω} {| \sum_{j = 1}^{k} b_{j} w_{j} |}^{q} d x + α_{0} C_{2} \int_{Ω} {| \sum_{j = 1}^{k} b_{j} w_{j} |}^{2} dx . \end{array}

Since

2 < q < p_{s}^{*}

, using the interpolation inequality (see [3, Theorem 2.11]) and (2.1), we get

(3.4)

\begin{array}{l} \int_{Ω} {| \sum_{j = 1}^{k} b_{j} w_{j} |}^{q} dx & \leq {‖ \sum_{j = 1}^{k} b_{j} w_{j} ‖}_{L^{2} (Ω; ℝ^{m})}^{θ q} {‖ \sum_{j = 1}^{k} b_{j} w_{j} ‖}_{L^{p_{s}^{*}} (Ω; ℝ^{m})}^{(1 - θ) q} \\ \leq C_{p_{s}^{*}} {‖ \sum_{j = 1}^{k} b_{j} w_{j} ‖}_{L^{2} (Ω; ℝ^{m})}^{θ q} {‖ \sum_{j = 1}^{k} b_{j} w_{j} ‖}_{W_{0}}^{(1 - θ) q}, \end{array}

where θ ∈ (0, 1) satisfies

\frac{1}{q} = \frac{θ}{2} + \frac{1 - θ}{p_{s}^{*}} \cdot

We observe that

(1 - θ) q = \frac{p_{s}^{*} (q - 2)}{p_{s}^{*} - 2} < p_{s}^{*}

and

λ : = \frac{p θ q}{p - (1 - θ) q} = \frac{2 p (p_{s}^{*} - q)}{p_{s}^{*} (p - q + 2) - 2 p} > 2 .

For any ϵ ∈ (0, 1), the Young inequality implies

(3.5)

\begin{array}{l} {‖ \sum_{j = 1}^{k} b_{j} w_{j} ‖}_{L^{2} (Ω; ℝ^{m})}^{θ q} {‖ \sum_{j = 1}^{k} b_{j} w_{j} ‖}_{W_{0}}^{(1 - θ) q} \\ \leq ɛ {‖ \sum_{j = 1}^{k} b_{j} w_{j} ‖}_{W_{0}}^{p} + C (ɛ) {‖ \sum_{j = 1}^{k} b_{j} w_{j} ‖}_{L^{2} (Ω; ℝ^{m})}^{λ} . \end{array}

Then, (3.4) is transformed into the following inequality

(3.6)

{\int_{Ω} | \sum_{j = 1}^{k} b_{j} w_{j} |}^{q} dx \leq C_{p_{s}^{*}} ɛ {‖ \sum_{j = 1}^{k} b_{j} w_{j} ‖}_{W_{0}}^{p} + C (ɛ) {‖ \sum_{j = 1}^{k} b_{j} w_{j} ‖}_{L^{2} (Ω; ℝ^{m})}^{λ} .

Plugging inequalities (3.3), (3.4) and (3.6) into (3.2), we deduce that

\begin{array}{l} \frac{1}{2} \frac{d | b (t) |^{2}}{dt} + {‖ \sum_{j = 1}^{k} b_{j} w_{j} ‖}_{W_{0}}^{p} \leq C_{p_{s}^{*}} α_{0} ɛ {‖ \sum_{j = 1}^{k} b_{j} w_{j} ‖}_{W_{0}}^{p} \\ + α_{0} C (ɛ) {‖ \sum_{j = 1}^{k} b_{j} w_{j} ‖}_{L^{2} (Ω; ℝ^{m})}^{λ} + α_{0} {‖ \sum_{j = 1}^{k} b_{j} w_{j} ‖}_{L^{2} (Ω; ℝ^{m})}^{2} . \end{array}

By choosing

ɛ = \frac{1}{2 α_{0} C_{p_{s}^{*}}}

, we get

(3.7)

\begin{array}{l} \frac{1}{2} \frac{d | b (t) |^{2}}{dt} + \frac{1}{2} {‖ \sum_{j = 1}^{k} b_{j} w_{j} ‖}_{W_{0}}^{p} \leq α_{0} C (ɛ) {‖ \sum_{j = 1}^{k} b_{j} w_{j} ‖}_{L^{2} (Ω; ℝ^{m})}^{λ} \\ + α_{0} {‖ \sum_{j = 1}^{k} b_{j} w_{j} ‖}_{L^{2} (Ω; ℝ^{m})}^{2} . \end{array}

It follows that

\frac{d | b (t) |^{2}}{dt} \leq 2 C_{3} ({‖ \sum_{j = 1}^{k} b_{j} w_{j} ‖}_{L^{2} (Ω; ℝ^{m})}^{λ} + {‖ \sum_{j = 1}^{k} b_{j} w_{j} ‖}_{L^{2} (Ω; ℝ^{m})}^{2}) .

Denote z(t) = |b(t)|², then

(3.8)

\frac{dz (t)}{dt} \leq 2 C_{3} (z {(t)}^{\frac{λ}{2}} + z (t)) .

Integrating (3.8) from 0 to t, and using the property

z (0) = | b (0) |^{2} = \int_{Ω} ξ_{k}^{2} (x) d x \leq C_{4},

we can conclude that

\begin{matrix} z (t) \leq exp (2 C_{3} t) {(C_{4}^{1 - \frac{λ}{2}} - exp (C_{3} (λ - 2) t))}^{\frac{2}{2 - λ}}, & as t < \frac{ln (C_{4}^{1 - \frac{λ}{2}})}{C_{3} (λ - 2)} . \end{matrix}

For

0 < T < T_{0} = \frac{ln (C_{4}^{1 - \frac{λ}{2}})}{C_{3} (λ - 2)}

, we obtain that |b(t)| ≤ C(T) ∀t ∈ [0, T ], where

C (T) = exp (2 C_{3} T) {(C_{4}^{1 - \frac{λ}{2}} - exp (C_{3} (λ - 2) T))}^{\frac{2}{2 - λ}} .

Put

\begin{matrix} J_{k} = max_{(t, b) \in [0, T] \times B (b (0), 2 C (T))} | S_{k} - R_{k} (t, b) | & and & β_{k} = min {T, \frac{2 C (T)}{J_{k}}} \end{matrix},

where B(b(0), 2C(T)) is the ball of center b(0) and radius 2C(T). By [16, Peano theorem], we know that problem (3.1) has a C¹ solution on [0, β_k]. Let b (β_k) be an initial value, then we can repeat the above process and get a C¹ solution on [β_k, 2β_k]. Without loss of generality, we assume that

\begin{matrix} T = [\frac{T}{β_{k}}] β_{k} + (\frac{T}{β_{k}}) β_{k}, & 0 < (\frac{T}{β_{k}}) < 1, \end{matrix}

where

[\frac{T}{β_{k}}]

is the integer part of

\frac{T}{β_{k}}

and

(\frac{T}{β_{k}})

is the decimal part of

\frac{T}{β_{k}}

. We can divide [0, T ] into [(i − 1)β_k, iβ_k], i = 1, . . . , N and [Nβ_k, T ] where

N = [\frac{T}{β_{k}}]

, then there exist C¹ solution

b_{k}^{i} (t)

in [(i − 1)β_k, iβ_k], i = 1, . . . , N and

b_{k}^{N + 1} (t)

in [Nβ_k, T ]. Therefore, we get a solution b_k(t) ∈ C¹([0, T ]) defined by

b_{k} (t) = {\begin{array}{l} b_{k}^{1} (t), & if t \in [0, β_{k}], \\ b_{k}^{2} (t), & if t \in (β_{k}, 2 β_{k}], \\ ⋮ & ⋮ \\ b_{k}^{N} (t), & if t \in ((N - 1) β_{k}, N β_{k}], \\ b_{k}^{N + 1} (t), & if t \in (N β_{k}, T] . \end{array}

As a result, we get the desired Galerkin approximation solution.

Assertion 2: A priori estimates

By (3.1), we have

(3.9)

\begin{array}{l} \int_{Ω} \frac{\partial u_{k}}{\partial t} w_{i} dx \\ + \iint_{Q_{Ω}} \frac{| u_{k} (x, t) - u_{k} (y, t) |^{p - 2} (u_{k} (x, t) - u_{k} (y, t))}{| x - y |^{n + p s}} (w_{i} (x) - w_{i} (y)) dxdy \\ = \int_{Ω} f (x, t, u_{k}) w_{i} dx, \end{array}

where 1 ≤ i ≤ k and t ∈ [0, T ] (T < T₀).

Multiplying (3.9) by (b(t))_i (resp. by $\frac{d}{dt} {(b (t))}_{i}$ ) and summing with respect to i from 1 to k, we arrive at (integrating with respect to t from 0 to τ (τ ∈ (0, T ]))

(3.10)

\begin{array}{l} \int_{Q_{τ}} \frac{\partial u_{k}}{\partial t} u_{k} dxdt + \int_{0}^{τ} {‖ u_{k} (x, t) ‖}_{W_{0}}^{p} dt = \int_{Q_{τ}} f (x, t, u_{k}) u_{x} dxdt, \\ \int_{Ω} {| \frac{\partial u_{k}}{\partial t} |}^{2} dx \\ + \iint_{Q_{Ω}} \frac{| u_{k} (x, t) - u_{k} (y, t) |^{p - 2} (u_{k} (x, t) - u_{k} (y, t))}{| x - y |^{n + p s}} (\frac{\partial u_{k} (x, t)}{\partial t} - \frac{\partial u_{k} (y, t)}{\partial t}) dxdy \\ = \int_{Ω} f (x, t, u_{k}) \frac{\partial u_{k}}{\partial t} dx . \end{array}

According to (3.7), we have

\frac{1}{2} \frac{d}{dt} \int_{Ω} u_{k} {(x, t)}^{2} dx + \frac{1}{2} {‖ u_{k} (x, t) ‖}_{W_{0}}^{p} \leq C_{5} ({(\int_{Ω} {| u_{k} |}^{2} dx)}^{λ / 2} + \int_{Ω} {| u_{k} |}^{2} dx) .

Similar to the estimation of b(t), we have

(3.11)

\begin{matrix} {\int_{Ω} | u_{k} (x, t) |}^{2} dx \leq C (T) & \forall t \in [0, T] & (T < T_{0}) \end{matrix} .

Moreover

(3.12)

{‖ u_{k} ‖}_{L^{p} (0, T; W_{0})} \leq C_{6} .

Hence, we get

(3.13)

{‖ u_{k} ‖}_{L^{\infty} (0, T; L^{2} (Ω; ℝ^{m}))} \leq C_{7} .

According to (3.10) and (H1), we get

(3.14)

\begin{array}{l} \int_{Ω} {| \frac{\partial u_{k}}{\partial t} |}^{2} dx \\ + \iint_{Q_{Ω}} \frac{| u_{k} (x, t) - u_{k} (y, t) |^{p - 2} (u_{k} (x, t) - u_{k} (y, t))}{| x - y |^{n + p s}} (\frac{\partial u_{k} (x, t)}{\partial t} - \frac{\partial u_{k} (y, t)}{\partial t}) dxdy \\ - \frac{d}{dt} \int_{Ω} F (x, t, u_{k}) dx \leq \int_{Ω} F_{t} (x, t, u_{k}) dx \leq α_{1} \int_{Ω} | u_{k} |^{q} dx + α_{1} . \end{array}

From the fact

\begin{array}{l} \frac{1}{p} \frac{d}{dt} {‖ u_{k} (x, t) ‖}_{W_{0}}^{p} \\ = \iint_{Q_{Ω}} \frac{| u_{k} (x, t) - u_{k} (y, t) |^{p - 2} (u_{k} (x, t) - u_{k} (y, t))}{| x - y |^{n + p s}} (\frac{\partial u_{k} (x, t)}{\partial t} - \frac{\partial u_{k} (y, t)}{\partial t}) dxdy, \end{array}

applied to (3.14), we deduce

(3.15)

\begin{array}{l} \int_{Ω} {| \frac{\partial u_{k}}{\partial t} |}^{2} dx + \frac{d}{dt} (\frac{1}{p} {‖ u_{k} (x, t) ‖}_{W_{0}}^{p} - \int_{Ω} F (x, t, u_{k}) dx) \\ \leq α_{1} (\int_{Ω} | u_{k} |^{q} dx + 1) . \end{array}

By using the same technique in (3.5) and using (3.11) to the term in the right-hand side of (3.15), we get

(3.16)

\begin{array}{l} \int_{Ω} {| \frac{\partial u_{k}}{\partial t} |}^{2} dx & + \frac{d}{dt} (\frac{1}{p} {‖ u_{k} (x, t) ‖}_{W_{0}}^{p} - \int_{Ω} F (x, t, u_{k}) dx) \\ \leq α_{1} ɛ C_{p_{s}^{*}} {‖ u_{k} (x, t) ‖}_{W_{0}}^{p} + α_{1} C (ɛ) {(\int_{Ω} u_{k} |^{2} dx)}^{λ / 2} + α_{1} \\ \leq C_{8} ({‖ u_{k} (x, t) ‖}_{W_{0}}^{p} + 1) . \end{array}

Integrating (3.16) with respect to t from 0 to τ (τ ∈ (0, T ]) and using the strong convergence in u_k(x, 0) → u₀(x) in W₀, we get

(3.17)

\begin{array}{l} \int_{Q_{τ}} {| \frac{\partial u_{k}}{\partial t} |}^{2} dxdt + \frac{1}{p} {‖ u_{k} (x, τ) ‖}_{W_{0}}^{p} & \leq C_{9} (\int_{0}^{τ} {‖ u_{k} (x, t) ‖}_{W_{0}}^{p} dt + 1) \\ + \int_{Ω} F (x, τ, u_{k}) dx . \end{array}

By assumption (H1) and interpolation inequality used in (3.5), we get

(3.18)

\int_{Ω} F (x, τ, u_{k}) dx \leq α_{1} ɛ C_{p_{s}^{*}} {‖ u_{k} (x, τ) ‖}_{W_{0}}^{p} + α_{1} C (ɛ) {(\int_{Ω} {| u_{k} |}^{2} dx)}^{λ / 2} .

Plugging (3.18) in (3.17), we arrive at

\begin{array}{l} \int_{Q_{τ}} {| \frac{\partial u_{k}}{\partial t} |}^{2} dxdt + \frac{1}{p} {‖ u_{k} (x, τ) ‖}_{W_{0}}^{p} \leq C_{9} (\int_{0}^{τ} {‖ u_{k} (x, t) ‖}_{W_{0}}^{p} dt + 1) \\ + α_{1} ɛ C_{p_{s}^{*}} {‖ u_{k} (x, τ) ‖}_{W_{0}}^{p} + α_{1} C (ɛ) {(\int_{Ω} | u_{k} (x, τ) |^{2} dx)}^{λ / 2} . \end{array}

By choosing

ɛ = \frac{1}{2 α_{1} p C_{p_{s}^{*}}}

, we get

\int_{Q_{τ}} {| \frac{\partial u_{k}}{\partial t} |}^{2} dxdt + \frac{1}{2 p} {‖ u_{k} (x, τ) ‖}_{W_{0}}^{p} \leq C_{10} (\int_{0}^{τ} {‖ u_{k} (x, τ) ‖}_{W_{0}}^{p} dt + 1) .

The Gronwall inequality implies that

\int_{0}^{τ} {‖ u_{k} (x, t) ‖}_{W_{0}}^{p} dt \leq C_{11}

for each τ ∈ [0, T ]. Therefore

\int_{Q_{τ}} {| \frac{\partial u_{k}}{\partial t} |}^{2} dxdt + \frac{1}{2 p} {‖ u_{k} (x, τ) ‖}_{W_{0}}^{p} \leq C_{12} .

We finally get

(3.19)

{\frac{\partial u_{k}}{\partial t}‖}_{L^{2} (Q_{T})} + {u_{k}‖}_{L^{\infty} (0, T; W_{0})} \leq C_{13} .

The assumption (H1) implies that

(3.20)

{f (x, t, u_{k})‖}_{L^{q^{'}} (Q_{T})} \leq C_{14} .

Assertion 3: Passage to the limit

By virtue of (3.12), (3.13), (3.19), and (3.20), we get the existence of a subsequence of (u_k) still denoted by (u_k) such that

(3.21)

\begin{array}{l} u_{k} ⇀ * u in L^{\infty} (0, T; L^{2} (Ω; ℝ^{m})) \cap L^{\infty} (0, T; W_{0}), \\ u_{k} ⇀ u in L^{p} (0, T; W_{0}), \\ \frac{\partial u_{k}}{\partial t} ⇀ \frac{\partial u}{\partial t} in L^{2} (Q_{T}; ℝ^{m}), \\ f (x, t, u_{k}) ⇀ χ in L^{q^{'}} (Q_{T}, ℝ^{m}) . \end{array}

[28, Theorem 5.1] and (3.21) imply that u_k → u in L^p(0, T, L²(Ω; ℝ^m)) and a.e. on Q_T (for a subsequence), and [28, Lemma 1.3] implies that f(x, t, u) = χ. We can conclude from the continuity in (H1),

f (x, t, u_{k}) u_{k} \to f (x, t, u) u a . e . in Q_{T} .

Using the Vitali Theorem, we get

lim_{k \to \infty} \int_{Q_{T}} f (x, t, u_{k}) u_{k} dxdt = \int_{Q_{T}} f (x, t, u) u dxdt .

\int_{Ω} u_{k} {(x, T)}^{2} dx \leq C_{15}

, we get the existence of a subsequence of (u_k) still denoted by (u_k) and a function û in L² (Ω; ℝ^m) such that u_k(x, T) → û in L² (Ω; ℝ^m). Then, for any b(t) ∈ C¹([0, T ]) and

ϕ \in C_{0}^{\infty} (Ω)

\int_{Q} \frac{\partial u_{k}}{\partial t} b ϕ dxdt = \int_{Ω} u_{k} (x, T) b (T) ϕ dx - \int_{Ω} u_{k} (x, 0) b (0) ϕ dx - \int_{Q} u_{k} \frac{\partial b}{\partial t} ϕ dxdt .

Tending k to ∞, we get

\int_{Ω} (\hat{u} - u (x, T)) b (T) ϕ dx - \int_{Ω} (u_{0} (x) - u (x, 0)) b (0) ϕ dx = 0 .

Choosing b(T) = 1, b(0) = 0 or b(T) = 0, b(0) = 1, we have û = u(x, T) and u₀(x) = u(x, 0).

As stated in the introduction, Young measure is the tool we use to prove the existence of a weak solution. To identify the weak limit, we consider the following lemma:

Lemma 3.4

Suppose that (3.12) holds. Then, the Young measure µ_(x,y,t) generated by $\frac{u_{k} (x, t) - u_{k} (y, t)}{{x - y|}^{\frac{n}{p} + s}} \in L^{p} (Q_{Ω} \times (0, T); ℝ^{m})$ has the following properties:

(a) ∥µ_(x,y,t)∥_ℳℝ^m= 1 for a.e. (x, y, t) ∈ Q_Ω × (0, T), i.e. µ_(x,y,t) is a probability measure.
(b) $μ_{(x, y, t)}, i d〉 = \int_{ℝ^{m}} λ d μ_{(x, y, t)} (λ)$ is the weak L¹-limit of $\frac{u_{k} (x, t) - u_{k} (y, t)}{{x - y|}^{\frac{n}{p} + s}}$ .
(c) $μ_{(x, y, t)}, i d〉 = \frac{u (x, t) - u (y, t)}{{x - y|}^{\frac{n}{p} + s}}$ for a.e. (x, y, t) ∈ Q_Ω × (0, T).

Proof

(a) For simplicity reasons, we consider
(3.22)
$v_{k} (x, y, t) = \frac{u_{k} (x, t) - u_{k} (y, t)}{{x - y|}^{\frac{n}{p} + s}} \in L^{p} (Q_{Ω} \times (0, T); ℝ^{m}) .$
We know that for any M > 0, (Ω ∩ B_M)² ⊆ Ω × Ω ⫅̸ Q_Ω, where B_M is the ball centered in 0 with radius M. Let N ∈ ℝ be such that
$Q_{N} \equiv (x, y, t) \in Ω \cap B_{M} \times Ω \cap B_{M} \times (0, T) : v_{k} (x, y, t)| \geq N\} .$
Using (3.12), we get
$\begin{array}{l} {v_{k}‖}_{L^{p} (Q_{Ω} \times (0, T); ℝ^{m})} & = {(\int_{0}^{T} \iint_{Q_{Ω}} \frac{{u_{k} (x, t) - u_{k} (y, t)|}^{p}}{{x - y|}^{n + p s}} dxdydt)}^{1 / p} \\ = {u_{k}‖}_{L^{p} (0, T; W_{0})} \leq M . \end{array}$
Consequently, there exists C₁₆ ≥ 0 such that
(3.23)
$C_{16} \geq \iint_{Q_{Ω} \times (0, T)} {v_{k} (x, y, t)|}^{p} dxdy \geq \iint_{Q_{N}} {v_{k} (x, y, t)|}^{p} dxdy \geq N^{p} Q_{N}|,$
where |Q_N | is the Lebesgue measure of Q_N. According to (3.23), the sequence (v_k) satisfies (2.2). Hence, a Young measure noted by µ_{(x, y, t)} is generated by v_k such that ∥µ_{(x, y, t)}∥_ℳ(ℝ^m) = 1 for a.e. (x, y, t) ∈ Q_Ω × (0, T).
(b) By (3.12), there exists a subsequence still denoted by (v_k) that converges in L^p (Q_Ω × (0, T); ℝ^m). Since L^p (Q_Ω × (0, T); ℝ^m) is reflexive, then v_k is weakly convergent in L¹ (Q_Ω × (0, T); ℝ^m). By the third assertion in Lemma 2.4, we replace the function ρ by the identity function, to obtain
$v_{k} ⇀ μ_{(x, y, t)}, i d〉 = \int_{ℝ^{m}} λ d μ_{(x, y, t)} (λ) weakly in L^{1} (Q_{Ω} \times (0, T); ℝ^{m}) .$
(c) According to (3.12), v_k is bounded in L^p (Q_Ω × (0, T); ℝ^m), then there exists a subsequence such that v_k ⇀ v in L^p (Q_Ω × (0, T); ℝ^m). Owing to the previous arguments, we get from the uniqueness of limits that
$μ_{(x, y, t)}, i d〉 = v (x, y, t) = \frac{u (x, t) - u (y, t)}{{x - y|}^{\frac{n}{p} + s}} for a . e . (x, y, t) \in Q_{Ω} \times (0, T) .$

Now, let {v_k} be the sequence given in (3.22), i.e.

v_{k} (x, y, t) = \frac{u_{k} (x, t) - u_{k} (y, t)}{{x - y|}^{\frac{n + p s}{p}}} .

The weak convergence given in Lemma 3.4 shows that

(3.24)

\begin{array}{l} {v_{k} (x, y, t)|}^{p - 2} v_{k} (x, y, t) & ⇀ \int_{ℝ^{m}} {λ|}^{p - 2} λ d μ_{(x, y, t)} (λ) \\ = {v (x, y, t)|}^{p - 2} v (x, y, t) \\ = \frac{{u (x, t) - u (y, t)|}^{p - 2} (u (x, t) - u (y, t))}{{x - y|}^{\frac{n + p s}{p^{'}}}} \end{array}

weakly in L¹ (Q_Ω × (0, T); ℝ^m). Since the space L^p is reflexive and |v_k(x, y, t)|^p−²v_k(x, y, t) is bounded in L^p′(Q_Ω × (0, T); ℝ^m), the sequence |v_k(x, y, t)|^p−²v_k(x, y, t) converges in L^p′ (Q_Ω × (0, T); ℝ^m). Hence its weak L^p′-limit is also |v(x, y, t)|^p−²v(x, y, t). Thus, for any φ ∈ L^p(0, T; W₀) we have

\frac{φ (x, t) - φ (y, t)}{{x - y|}^{\frac{n + p s}{p}}} \in L^{p} (Q_{Ω} \times (0, T); ℝ^{m}) .

According to the weak limit in (3.24), we get

\begin{array}{l} lim_{k \to \infty} \int_{0}^{T} \iint_{Q_{Ω}} \frac{{u_{k} (x, t) - u_{k} (y, t)|}^{p - 2} (u_{k} (x, t) - u_{k} (y, t))}{{x - y|}^{n + p s}} (φ (x, t) - φ (y, t)) dxdydt \\ = \int_{0}^{T} \iint_{Q_{Ω}} \frac{{u (x, t) - u (y, t)|}^{p - 2} (u (x, t) - u (y, t))}{{x - y|}^{n + p s}} (φ (x, t) - φ (y, t)) dxdydt \end{array}

for every φ ∈ L^p(0, T; W₀).

From (3.9), for ϕ ∈ C¹ (0, T; U_M), M ≤ k, we have

\begin{array}{l} \int_{Q_{T}} \frac{\partial u_{k}}{\partial t} ϕ dxdt \\ + \int_{0}^{T} \iint_{Q_{Ω}} \frac{{u_{k} (x, t) - u_{k} (y, t)|}^{p - 2} (u_{k} (x, t) - u_{k} (y, t))}{{x - y|}^{n + p s}} (ϕ (x, t) - ϕ (y, t)) dxdydt \\ = \int_{Q_{T}} f (x, t, u_{k}) ϕ dxdt . \end{array}

For k tending to ∞, it follows from the above results, that

(3.25)

\begin{array}{l} \int_{Q_{T}} \frac{\partial u}{\partial t} ϕ dxdt \\ + \int_{0}^{T} \iint_{Q_{Ω}} \frac{{u (x, t) - u (y, t)|}^{p - 2} (u (x, t) - u (y, t))}{{x - y|}^{n + p s}} (ϕ (x, t) - ϕ (y, t)) dxdydt \\ = \int_{Q_{T}} f (x, t, u) ϕ dxdt . \end{array}

for all

ϕ \in C^{1} (0, T; \underset{M \geq 1}{\cup} U_{M})

. Letting M goes to infnity, consequently, (3.25) holds for all

ϕ \in C^{1} (0, T; C_{0}^{\infty} (Ω))

4. An example

We consider the following problem

\begin{array}{l} \frac{\partial u}{\partial t} + {(- Δ)}_{p}^{s} u = a (x, t) | u |^{q - 2} u & in Q_{T} = Ω \times (0, T), \\ u = 0 & in Cscr Ω \times (0, T), \\ u (x, 0) = u_{0} (x) & in Ω, \end{array}

comparing it with problem (1.1) where f(x, t, u) = a(x, t) |u|^q−2u,

F (x, t, u) = \frac{a (x, t)}{q} {u|}^{q}

, and F_t(x, t, u) ⩾ C(−|r|^q − 1). If

2 < q < p_{s}^{*}

, then by Theorem 3.2, there exists a constant T₀ > 0 such tha the problem (1.1) has a weak solutions as T < T₀.

On Weak Solutions to Parabolic Problem Involving the Fractional p-Laplacian via Young Measures

Full Article

1. Introduction

(1.1)

2. Preliminaries and notations

(2.1)

Lemma 2.1. ([20])

Lemma 2.2. ([18])

Definition 2.3. ([8])

Lemma 2.4. ([23])

(2.2)

3. Local existence of weak solutions

Definition 3.1

Theorem 3.2

Proof

Assertion 1: Galerkin approximation

Lemma 3.3

Proof

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

Assertion 2: A priori estimates

(3.9)

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

(3.18)

(3.19)

(3.20)

Assertion 3: Passage to the limit

(3.21)

Lemma 3.4

Proof

(3.22)

(3.23)

(3.24)

(3.25)

4. An example