Abstract
In this work, we study the existence of a capacity solution for a nonlocal thermistor problem in Musielak–Orlicz–Sobolev spaces. We get the existence of capacity solution using the approximate techniques and we prove the existence of a weak solution by introducing a sequence of approximate problems converging in a certain sense to a capacity solution. As a consequence, we obtain the existence of a capacity solution of the original problem in Musielak–Orlicz–Sobolev Lebesgue spaces.
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1 Introduction
In recent decades, Sobolev spaces and Musielak–Orlicz spaces have become of great interest in the study of different problems [10]. In the context of Musielak–Orlicz spaces, the first work was done by Orlicz in 1930, followed by the work of Nakano in 1950 [24], in which the author presented a general study of these spaces. On the other hand, Czechoslovak and Polich investigated the modular function spaces. When the Leray Lions operator satisfies the nonpolynomial growth condition, the study of variational problems becomes more interesting in the applications to electro-rheology. Ruzicka and Rajagopal proposed a mathematical model of electro-rheological fluids (see [27, 28] for more details).
We consider the following problem modeling the temperature produced by a material crossed by an electric current flow:
where f(u) is the electrical resistance of the conductor and \(\displaystyle {\frac{f(u)}{(\int _{\Omega }f(u)\mathrm{{d}}x)^{2}}}\) represents the nonlocal term of (1.1). Whereas \(Q_S\) is defined as follows \(Q_S:=\Omega \times [0,S]\) where \(\Omega \) is an open-bounded subset of \(\mathbb {R}^N,~N\ge 1\), S is a positive constant, and ]0, S[ denotes the time horizon. The first equation in problem (1.1) describes the diffusion of temperature in the presence of a nonlocal term as a consequence of Joule effect in the second member. \(\lambda \) is a constant without dimension and can be identified by the square of the applied potential difference at the ends of the conductor. The function u represents the temperature generated by the electric current flowing through the material [9, 22]. There are various motivations behind the analysis of the heat and current flow in thermistors. One is the obvious question of design: how do the characteristics, such as the switch-off time in response to a current surge, depend on the physical parameters. Another is an issue of quality control: some thermistors can crack, because rapid thermal expansion caused by large temperature gradients stresses the material too much. In 1833, Micheal Faraday (1791–1867) discovered the thermistor and remarked that the augmentation of temperature implies a decrease of the Silver Sulfides resistance. The thermistor is defined as a temperature sensing device.
The thermistor problem has been excessively used by many authors (Antontsev and Chipot 1999; [4]; Gonzàlez Montesinos and Ortégeon Gallgo 2002; Kutluay and Esen 2005; [13] ). They proved the existence of solution for thermistor problem in the context of vector-valued Sobolev spaces or in the standard Sobolev spaces. Hence, it is interesting to develop and analyze thermistor problem in the context of Musielak–Orlicz–Sobolev spaces.
Our aim is to prove the existence of a capacity solution in the sense of Definition 4.2 to system (1.1) which is the transformation of a coupled system consisting of an elliptic equation describing the quasistatic evolution of the electric potential and a nonlinear parabolic equation, which describes the temperature [22]. The literature on problems (1.1) and coupled system recall above is vast (see [1, 7, 14,15,16, 20, 25, 29]).
The rest of this paper is organized as follows: in Sect. 2, we state some basic concepts and a few known results that are useful for the results that will be established in this paper. In Sect. 3, we give the compactness results and the assumptions on data. In Sect. 4, we introduce the concept of capacity solutions. In Sect. 5, we develop the state of the main result of this paper. Finally, we give a conclusion and some perspectives.
2 Preliminaries
In this section, we introduce some definitions, proprieties, and basic notions of this work needed in the next sections.
Definition 2.1
(See [10]) Let \(\psi :\Omega \times \mathbb {R}\longrightarrow \mathbb {R}\), then \(\psi \) is Musielak–Orlicz functions if it satisfies two following conditions:
-
1.
\(\psi (.,s)\) is a measurable function for all s in \(\mathbb {R}\).
-
2.
For each x in \(\Omega \), we have \(\psi (x,.)\) is a N-function; also, it is convex in \(\mathbb {R}\) and increasing in \(\mathbb {R}^{+}\), such that
$$\begin{aligned} \begin{aligned}&\lim \limits _{s\longrightarrow 0}\dfrac{\psi (x,s)}{s}=0,~ \lim \limits _{s\longrightarrow \infty }\dfrac{\psi (x,s)}{s}=\infty , \\&\psi (x,s)>0, \text{ for } \text{ all } s>0,\\&\psi (x,s)=0, \text{ for } s=0. \end{aligned} \end{aligned}$$
Definition 2.2
(See [23]) Let \(\phi \) and \(\psi \) be two Musielak–Orlicz functions defined in \(\Omega \times \mathbb {R}\) with values in \(\mathbb {R}\), then \(\psi \) dominates \(\phi \) globally (\(\phi<<\psi \)) if \(\exists r>0\) and \(\exists s_0\ge 0\), such that
we also say \(\psi \) dominates \(\phi \) globally if \(s_0=0\) and beside infinity if \(s_0>0.\)
We define the space
where \({\varrho _{\psi ,\Omega }(u)=\int _{\Omega }\psi ( x,u( x))\mathrm{{d}}x}.\)
Let \(L_\psi (\Omega )\) the Musielak–Orlicz space generated by \(\displaystyle {F_{\psi }(\Omega )} \), such that this last space is the Musielak–Orlicz class and it is the smallest vector space of the following space:
We define the complementary function of the Musielak function \(\psi (x,r)\) in the sense of Young with respect to variable t as follows:
Then, Young Fenchel inequality is defined by
We endow the space \(L_\psi (\Omega )\) by Luxemburg norm
or by Orlicz norm
Moreover, the following inequality holds:
Using the above inequality, we get
Also, we have the equivalent between Luxemburg norm and Orlicz norm
For the proof, we refer to [23]. We also have the Hölder’s inequality holds
if \(\Omega \) has a finite measure, the inequality (2.4) implies the following continuous inclusion: \(L_\psi (\Omega )\subset L^1 (\Omega )\) which is strict in general.
We denote by \(E_\psi (\Omega )\) the set of the closure of bounded measurable functions with compact support in the closure of \(\Omega \) denoted by \(\overline{\Omega }\) with respect to the norm of \(L_\psi (\Omega ).\)
Throughout this paper, we will use the standard reference for Musielak–Orlicz–Sobolev spaces [23]; see also [3]. Now, we introduce some definition and lemmas useful hereafter.
Definition 2.3
Let \(\displaystyle {(u_n)_{n\in \mathbb {N}}\subset L_\psi (\Omega )}\), we say \(\displaystyle {(u_n)_{n\in \mathbb {N}}}\) converges to \(\displaystyle {u\in L_\psi (\Omega )}\) if there exists \(\displaystyle { l>0}\), such that
For all \(p\in \mathbb {N}\), we defined a Musielak–Orlicz–Sobolev spaces as follows:
where \(\alpha =(\alpha _1,\alpha _2,\alpha _3,...,\alpha _{m-1},\alpha _m)\in \mathbb {Z}^m\), \(\mid \alpha \mid =\alpha _1+\alpha _2+\alpha _3+ \cdots +\alpha _{m-1}+\alpha _m\) and
\(D^{\alpha }\) is the distributional derivative of multi-index \(\alpha \).
For each Musielak–Orlicz–Sobolev space \(W^pL_\psi (\Omega )\), we define the modular as follows:
which is convex in \(W^pL_\psi (\Omega ).\) We can equipped Musielak–Orlicz–Sobolev space with
The above two norms are equivalent on \(W^pL_\psi (\Omega )\). The pair \(\displaystyle {(W^pL_\psi (\Omega ),\parallel u\parallel ^{(p)}_{\psi ,\Omega })}\) is a Banach space [23], if \(\exists z_0>0\), such that
Then, \(\displaystyle {(W^pL_\psi (\Omega ),\parallel u\parallel _{p,\psi ,\Omega })}\) is a Banach space.
Hereafter, we suppose that the condition (2.5) is satisfied. The space \(\displaystyle {W^pL_\psi (\Omega )}\) can be identified to a \(\sigma (\Pi _{\mid \alpha \mid \le p}L_\psi (\Omega ),\Pi _{\mid \alpha \mid \le p}E_{\overline{\psi }}(\Omega ))\)-closed subspace of \(\Pi _{\mid \alpha \mid \le p}L_\psi (\Omega )\). Let \(\displaystyle {W^{p}_0L_\psi (\Omega )=\overline{D(\Omega )}^{\sigma (\Pi _{\mid \alpha \mid \le p}L_\psi (\Omega ),\Pi _{\mid \alpha \mid \le p}E_{\overline{\psi }}(\Omega ))}}\) and \(W^pE_\psi (\Omega )\) the spaces of functions u, where u and its distribution derivatives up to order m lie in \(E_\psi (\Omega )\). Moreover, \(W^p_0E_\psi (\Omega )\) is the norm closure of \(D(\Omega )\) in \(W^pL_\psi (\Omega )\).
Lemma 2.4
(Poincaré’s inequality see [2]) Let \(\Omega \) a subset of \(\mathbb {R}^N\) a bounded Lipchitz-continuous set, then there exists a constant \(C=C(\Omega )>0\), such that
Remark 2.5
Let \(u\in W^p_0L_\psi (\Omega )\) where \(\psi \) is a Musielak–Orlicz function, we suppose that there exists a positive constant C, such that
then
Using the convexity of \(\psi ( x,.)\) and if \(C\ge 1\), we get
if not, i.e., \(C<1\), we obtain \({ \int _{\Omega }\psi ( x,\nabla u)\mathrm{{d}}x\le C<1, \text{ then } \parallel \nabla u\parallel _{\psi ,\Omega }\le 1}.\)
In view of the fact that \(u\in W^p_0L_\psi (\Omega )\), we apply Lemma 2.4, we get that there exists a positive constant \(C=C(\Omega )\), such that
On the other hand, we have \(\displaystyle {\parallel u\parallel _{1,\psi ,\Omega }=\parallel u\parallel _{\psi ,\Omega }+\parallel \nabla u\parallel _{\psi ,\Omega }},\) and hence
Then
In the next of this paper, we suppose that \(\psi \) and \(\phi \) are two generalized N-function, such that \(\psi<<\phi \). We also assume that the following conditions hold for complementary functions \(\overline{\psi }\) and \(\overline{\phi }\)
Remark 2.6
(See [18], Remark 2.1) We suppose (2.7) and (2.8) hold, then
Definition 2.7
Let \(\displaystyle {(u_n)_{n\in \mathbb {N}}\subset W^pL_\psi (\Omega ) },\) we say that \(\displaystyle {(u_n)_{n\in \mathbb {N}}}\) converges to \(\displaystyle {u\in W^pL_psi(\Omega )}\) for the modular convergence in \(\displaystyle {W^pL_\psi (\Omega )}\) if and only if
Also, we can define these spaces of distributions as follows:
Lemma 2.8
Let \(\displaystyle {(u_n)_{n\in \mathbb {N}}\subset L_\psi (\Omega ) },\) If \(\phi<<\psi \) and \(\displaystyle {(u_n)_{n\in \mathbb {N}} }\) converges to \(\displaystyle {u\in L_\psi (\Omega ) }\), in the sense of modular convergent, then \(\displaystyle {(u_n)_{n\in \mathbb {N}} }\) converges to u strongly in \(\displaystyle {E_\phi (\Omega )}\). In particular, the following continuous injection hold: \(\displaystyle {L_\psi (\Omega )\subset E_\phi (\Omega )}\) and \(\displaystyle {L_{\overline{\phi }}(\Omega )\subset E_{\overline{\psi }}(\Omega )}\).
Proof
By hypothesis, for \(\ell >0\) and \(\varepsilon >0\), we have
Then, there exists \(h_0\in L^1(\Omega )\), such that
for a subsequence of \(\displaystyle {(u_n)_{n\in \mathbb {N}}}\) which is still denoted \(\displaystyle {(u_n)_{n\in \mathbb {N}}}\) for convenience. Knowing that \(\displaystyle {\phi<<\psi }\), then by applying Definition 2.2, there exists \(\displaystyle { k>0}\), such that \({\lim \nolimits _{ s\longrightarrow \infty }\sup \nolimits _{x\in \Omega }\dfrac{\phi (x,ks)}{\psi (x,s)}=0}\). As a consequence, there exists \(\displaystyle { s_0\ge 0}\), such that
Let set \(\displaystyle {k=\dfrac{\ell }{\varepsilon }}\) and \(s=\dfrac{t}{\ell }\) where \(t=u_n-u\), and hence
Using the characteristic function \(\chi _{\Omega }\), we get
Then
From Remark 2.1 in [18], we get \({ \sup \nolimits _{s\in [0, \ell s_0]}\mathrm{{ess}}\sup \nolimits _{x\in \Omega }\phi (x,s)<+\infty }.\)
Then, there exists \(h_1>0\), such that
Thanks to Lebesgue’s dominated convergence theorem, we have
for n near infinity, we obtain
The continuous embedding \(\displaystyle {L_\psi (\Omega )\subset E_\phi (\Omega )}\) is trivial, because the convergence in \(L_\psi (\Omega )\) implies the modular convergence. On the other side, we have \(\displaystyle {\phi<<\psi }\) is equivalent to \(\displaystyle {\overline{\psi }<<\overline{\phi }}\); as a consequence, the following embedding \(\displaystyle {L_{\overline{\phi }}(\Omega )\subset E_{\overline{\psi }}(\Omega )}\) is continuous. \(\square \)
Lemma 2.9
(See [12]) Let \(\displaystyle {(f_n)_{n\in \mathbb {N}}}\) and \(\displaystyle {(g_n)_{n\in \mathbb {N}}}\) be two convergent sequences in \(\displaystyle {L_\psi (\Omega )}\) and \(\displaystyle {L_{\overline{\psi }}(\Omega )}\), respectively, and denote by \(\displaystyle {f\in L_\psi (\Omega )} \) and \(\displaystyle {g\in L_{\overline{\psi }}(\Omega )}\) their corresponding limits in the sense of modular convergence, then
Lemma 2.10
(See [6]) Let \(\Omega \) be a bounded, Lipchitz-continuous subset of \(\mathbb {R}^N,\) \(\psi \) a Musielak–Orlicz function and \(\overline{\psi }\) its complementary. Then
-
\(D(\Omega )\) is dense in \(L_\psi (\Omega )\) with respect to the modular convergence.
-
\(D(\Omega )\) is dense in \( W^1_0L_\psi (\Omega )\) and \(D(\overline{\Omega })\) is dense in \( W^1L_\psi (\Omega ).\)
The previous densities are with respect to the modular convergence. Moreover, all the previous densities hold true if the following conditions are satisfied:
-
(1)
There exists a constant \(\lambda >0\), such that \(\forall x,y\in \Omega ,\) \(\mid x-y\mid \le \dfrac{1}{2}\) implies
$$\begin{aligned} \displaystyle {\dfrac{\phi (x,\ell )}{\phi (y,\ell )}\le ~\ell ^{-\dfrac{\lambda }{\log (\mid x-y\mid )}}~~\text{ for } \text{ all }~~\ell \ge 1.} \end{aligned}$$(2.11) -
(2)
There exists a constant \(\beta >0\), such that
$$\begin{aligned} \displaystyle {\overline{\psi }(x,1)\le \beta , \text{ a.e } \text{ in } \Omega }. \end{aligned}$$(2.12)
Remark 2.11
Define the measurable function \(q:\Omega \longrightarrow ]1,\infty [\) and suppose that there exists a positive constant C, such that for all \(x,y\in \Omega \) with \(\mid x-y\mid <\dfrac{1}{2}\), we have
Then, the following Musielak–Orlicz functions:
-
(1)
\(\psi (x,\ell )=\ell ^{q(x)}\),
-
(2)
\(\psi (x,\ell )=\ell ^{q(x)}\log (1+\ell )\),
-
(3)
\(\psi (x,\ell )=\ell \log (1+\ell )(\log (e-1+\ell ))^{q(x)},\)
satisfy the inequality (2.11).
Now, let us introduce inhomogeneous Musielak–Orlicz–Sobolev spaces. Let \(\Omega \subset \mathbb {R}^N\) be an open-bounded set and \(\psi \) a Musielak–Orlicz function defined in \(Q_S:=\Omega \times ]0.S[\) with \(S>0\). We denote by \(D^\alpha _x\) the distributional derivative on \(Q_S\) of order \(\alpha \in \mathbb {Z}^N\), where \(\alpha \) is a multi-index with respect to the variable x. We define the inhomogeneous Musielak–Orlicz–Sobolev spaces as follows:
we equip the spaces \(\displaystyle {W^{p,x}L_\psi (Q_S)}\) and \(\displaystyle {W^{p,x}E_\psi (Q_S)}\) with the norm
For \(p=1\), the pairs \(\displaystyle {(W^{p,x}L_\psi (Q_S),\parallel .\parallel )}\) and \(\displaystyle {(W^{p,x}E_\psi (Q_S),\parallel .\parallel )}\) are Banach spaces [17]. The two last spaces are considered as subspaces of the product space
We consider the weakly star topology \(\displaystyle {\sigma (\Pi _{\mid \alpha \mid \le p}L_\psi (Q_S),\Pi _{\mid \alpha \mid \le p}E_{\overline{\psi }}(Q_S))}\) and \(\displaystyle {\sigma (\Pi _{\mid \alpha \mid \le p}L_\psi (Q_S),\Pi _{\mid \alpha \mid \le p}L_{\overline{\psi }}(Q_S))}\). If \(\displaystyle {u\in W^{p,x}L_\psi (Q_S)}\), then the following mapping:
is well defined. Moreover, if \(\displaystyle {u\in W^{1,x}E_\psi (Q_S)},\) this function is a \(W^{1}E_\psi (\Omega )\)-valued function and is strongly measurable. We cannot assure the measurability of the function u(s) on ]0.S[. However, the function \(\displaystyle {s\longrightarrow \parallel u(s)\parallel _{\psi ,\Omega }}\) belongs to the space \(\displaystyle {L^1(]0.S[ )}\). We define the space \(\displaystyle {W^{1,x}_0E_\psi (Q_S)}\) as follows:
If \(\Omega \) is a Lipschitz-continuous domain, we can show as in [6] that each element u in the closure of \(D(Q_S)\) with respect of weak-\(*\) topology associated \(\displaystyle {(\sigma (\prod L_{\psi },\prod E_{\overline{\psi }}))} \) is a limit in \(\displaystyle {W^{1,x}L_\psi (Q_S)},\) of subsequence \(\displaystyle {(u_n)_{n\in \mathbb {N}}\subset D(Q_S)}.\) We emphasize that the modular convergence, i.e., there exists a positive constant \( \ell \), such that for all \(\mid \alpha \mid \le 1\), we have
implies that the sequence \(\displaystyle {(u_n)_{n\in \mathbb {N}}}\) converges to u in \(\displaystyle {W^{1,x}L_\psi (Q_S)}\) for the weak-\(*\) topology \(\displaystyle {\sigma (\prod L_{\psi },\prod L_{\overline{\psi }})} \). Consequently, we obtain
This space is denoted by \(\displaystyle {W^{1,x}_0E_\psi (Q_S)}\). Moreover, we have
In \(W^{1,x}_0L_\psi (Q_S)\), the following Poincaré’s inequality holds:
We denote by \(W^{-1,x}L_\psi (Q_S)\) the topologic dual of \(\displaystyle {W^{1,x}_0E_\psi (Q_S)}\) characterized by
which can be equipped by the usual quotient norm
Furthermore, we denote \(\displaystyle {W^{-1,x}E_{\overline{\psi }}(Q_S)}\) the subspace of \(\displaystyle {W^{-1,x}L_{\overline{\psi }}(Q_S)}\) consisting of linear forms which are \(\displaystyle {(\sigma (\prod L_{\psi },\prod E_{\overline{\psi }}))} \)-continuous. It can be shown that
In the sequel, we need the following lemma.
Lemma 2.12
We assume that \( \phi \) is a Musielak function verifying the condition (2.8) and we suppose that \(s^2\le \psi (x,s) \text{ for } \text{ all } x\in \Omega \text{ and } s\in \mathbb {R}\). Then, the following embedding:
are continuous. In particular, \( W^{1}_0L_\phi (\Omega )\hookrightarrow H^{1}_0(\Omega )\) and \(H^{-1}(\Omega )\hookrightarrow W^{-1}L_{\overline{\phi }}(\Omega ).\) Moreover, if \(\psi \) is a Musielak function verifying (2.7) and \(\phi<<\psi \), then the following embeddings:
are continuous. Consequently, the following embeddings \(\displaystyle { W^{1}_0L_\psi (\Omega )\hookrightarrow H^{1}_0(\Omega )}\) and \(\displaystyle {H^{-1}(\Omega )\hookrightarrow W^{-1}L_{\overline{\psi }}(\Omega )}\) are continuous.
Proof
By hypothesis, we have \(\displaystyle {v^2\le \phi (x,v)~~for~all~x\in \Omega }\), from whence follows that:
we set \(\displaystyle {v=\frac{u}{\parallel u\parallel _{(\phi ),\Omega }}}\) and \(\displaystyle {u\ne 0}\)
It yields that
which proves the first embedding.
Now, let \(\displaystyle {\phi<<\psi , }\) for \(\displaystyle {r\in ]0.S[}\)
Then, for \(\displaystyle {v\in F_{\psi }(\Omega )}\) and using Remark 2.1 in [18], we deduce the existence of a positive constant \(\displaystyle {C_1}\), such that
We replace \(\displaystyle {v}\) in the above inequality by \(\displaystyle {\frac{u}{\parallel u\parallel _{(\psi ),\Omega }}}\) where \(\displaystyle {u\ne 0}\), and we use (2.2) and we get
where \(\displaystyle {C_2=(C_1+r)^{1/2}}\). \(\square \)
Remark 2.13
We assume that the hypothesis of Lemma 2.12 is satisfied, then
For the proof, it suffices to assume that \(\displaystyle {g\in L^2(]0.S[;H^{-1}(\Omega ))}\), then for \(g_\alpha \in L^2(Q_S)\), \(g=\sum _{\vert \alpha \vert \le 1}D^\alpha _x g_{\alpha } \) and by Lemma 2.8, we have
and thus
We introduce the truncation operation \(S_{R}:\mathbb {R}\longrightarrow \mathbb {R}\), appearing in [8]
Then, its primitive is defined as follows:
3 Compactness results
In this section, we state trace and mollification results.
Let \(\Omega \) be an open-bounded subset of \(\mathbb {R}^N\) with a Lipschitz-continuous boundary, and \(\psi \) be a Musielak function. We set \(\displaystyle {Q_S =]0.S[\times \Omega }\). For \(\displaystyle {u\in L^1(Q_S)}\), \(\displaystyle {\eta >0}\), \(\displaystyle {r\in [0,S]}\) and \(\displaystyle {x\in \Omega }\), we define \(\displaystyle {u_\eta }\) as follows:
where \(\displaystyle {\widetilde{u}(x,t)=u(x,t)\chi _{]0,S[}}.\)
The following lemmas play a crucial role in the sequel of this paper.
Lemma 3.1
(See [11]) The following assertions hold:
-
(1)
Given any function \(\displaystyle {u\in L_\psi (Q_S)}\), then \(\displaystyle {u_\eta \in C([0,S]; L_\psi (\Omega ))}\) and \({\lim \nolimits _{\eta \longrightarrow \infty }u_\eta =u}\) in \(\displaystyle {L_\psi (Q_S)}\) for the modular convergence.
-
(2)
Let \(\displaystyle {u\in W^{1,x}L_\psi (Q_S)}\), we have \(\displaystyle {u_\eta \in C([0,S]; W^1L_\psi (\Omega ))}\) and \({\lim \nolimits _{\eta \longrightarrow \infty }u_\eta =u}\) in \(\displaystyle {W^{1,x}L_\psi (Q_S)}\) for the modular convergence.
-
(3)
Let \(\displaystyle {u\in E_{\psi }(Q_S)}\) (resp, \(\displaystyle {u\in W^{1,x}E_\psi (Q_S)}\)). \({\lim \nolimits _{\eta \longrightarrow \infty }u_\eta =u}\) strongly in \(\displaystyle {E_\psi (Q_S)}\) (resp, strongly in \(\displaystyle { W^{1,x}E_\psi (Q_S)}\)).
-
(4)
Let \(\displaystyle {u\in W^{1,x}L_\psi (Q_S)}\), then \(\dfrac{\partial u_\eta }{\partial s}=\eta (u-u_\eta )\in W^{1,x}L_\psi (Q_S).\)
-
(5)
Let \(\displaystyle {(u_n)_{n\in \mathbb {N}} }\) be a sequence in \(\displaystyle { W^{1,x}L_\psi (Q_S)}\) and \(\displaystyle {u\in W^{1,x}L_\psi (Q_S)}\), such that \(u_n\longrightarrow u\) as \(n\longrightarrow \infty \) strongly in \(\displaystyle { W^{1,x}L_\psi (Q_S)}\) (resp, for the modular convergence). Then, for each \(\eta >0\), we obtain \(\displaystyle {(u_n)_\eta \longrightarrow u_\eta }\) strongly in \(\displaystyle { W^{1,x}L_\psi (Q_S)}\) (resp, for the modular convergence).
Lemma 3.2
(See [11]) The following embedding:
is continuous.
Lemma 3.3
(See [11]) The following embeddings:
are continuous.
The Lemmas 3.4 and 3.6 play a key role in the proof of Theorem 5.2, whereas the Lemma 3.5 plays an important role in the Step III.
Lemma 3.4
(See [12]) Given a Banach space Y, such that \(L^1(\Omega )\hookrightarrow Y\) is a continuous embedding. If H is bounded in \(\displaystyle {W^{1,x}_0L_\psi (Q_S)}\) and relatively compact in \(L^1(0,S;Y)\), then H is relatively compact in \(L^1(Q_S)\) and in \(E_\phi (Q_S)\) for every \(\phi<<\psi \).
Lemma 3.5
(See [12]) Let \(\Omega \) be an open-bounded subset of \(\mathbb {R}^N\) with the segment property. Then, the following inclusion:
holds with a continuous embedding.
Lemma 3.6
(See Theorem 2 in [12]) Let \(\psi \) be a Musielak function. If H is a bounded subset of \(W_{0}^{1, x} L_{\psi }\left( Q_S\right) \) and \(\displaystyle {\lbrace \frac{\partial g}{\partial t} / g\in H \rbrace }\) is bounded in \(W^{-1, x} L_{\bar{\psi }}\left( Q_S\right) \), then H is relatively compact in \(L^{1}\left( Q_S\right) \).
In our study, we obtain the existence of a weak solution by applying Theorem 3.7.
We define the partial differential operator as follows:
\(\displaystyle {B: D(B)\subset W^{1, x} L_{\psi }\left( Q_S\right) \longrightarrow W^{1, x} L_{\psi }\left( Q_S\right) }\), such that \(B(u)=-\mathrm{{div}} ~a(x,s,\nabla u)\) where \(a(.,.,.):\Omega \times ]0.S[\times \mathbb {R}^N\longrightarrow \mathbb {R}^N\) is a Carathéodory function, and for all \((x,s)\in Q_S\), two real numbers \(\lambda >0\), \(k\ge 0\) and \(z,y \in \mathbb {R}^N\) where \(z\ne y\), the following conditions hold:
For each function \(g\in W^{-1, x} L_{\bar{\psi }}\left( Q_S\right) \) and for all \(u_0\in L^2 (\Omega ),\) we consider the following parabolic problem:
The following theorem plays a key role in the proof of Theorem 5.2.
Theorem 3.7
(See [26]) We suppose that the conditions (3.5) to (3.7) are satisfied, then the problem (3.8) has a weak solution \(u\in D(B)\cap W^{1, x}_0 L_{\psi }\left( Q_S\right) \cap C( [0,S];L^2(\Omega ))\) where \( a(x,s,\nabla u)\in W^{-1, x} L_{\bar{\psi }}\left( Q_S\right) \), and for all \(\omega \in W^{1, x}_0 L_{\psi }\left( Q_S\right) \) with \(\dfrac{\partial \omega }{\partial s}\in W^{-1, x} L_{\bar{\psi }}\left( Q_S\right) \) and for all \(r\in [0,S]\), we have
where \(\langle ., .\rangle _{Q_{r}}=\langle ., .\rangle _{ W^{-1, x} L_{\bar{\psi }}\left( Q_S\right) , W^{1, x}_0 L_{\psi }\left( Q_S\right) }.\) Moreover, for all \(r\in [0,S]\), the following energy identity holds:
4 Concept of capacity solution
Here, we define the concept of capacity solution for problem (1.1) in the context of the Musielak–Orlicz–Sobolev spaces. Now, let \(\Omega \subset \mathbb {R}^N\) be an open-bounded set and let \(\psi \) a Musielak function verifying the inequalities (2.11) and (2.12).
We equip the space \(\textit{F}\) by the following norm:
The pair \((\textit{F},\parallel . \parallel _{\textit{F}})\) is a Banach space.
In the sequel of this paper, we consider \(\langle ., .\rangle _{Q_{r}}=\langle ., .\rangle _{ W^{-1, x} L_{\bar{\psi }}\left( Q_S\right) , W^{1, x}_0 L_{\psi }\left( Q_S\right) },\) and we assume the following conditions:
Let \(\bar{\phi } \text{ and } \bar{\psi }\) be two complementary functions of the Musielak functions \(\phi (x,r)\) and \(\psi (x,r)\), respectively, satisfying the conditions (2.9) and (2.10), respectively. We consider also the operator
where \(Bu= -{\text {div}} a(x, s, \nabla u)\), such that a(., ., .) is a Leray–Lions operator where \(a(x,s,\nabla u)=\mid \nabla u\mid ^{p-2} \nabla u\). In our case, we take \(p=2\) where \(\displaystyle {\nabla : \mathbb {R}^N\longrightarrow \mathbb {R}^N}\) satisfies the following assumptions, for all \((x,s)\in Q_S\):
where \(\displaystyle {c(x,s)\in E_{\bar{\psi }}(Q_S)},\) \(\displaystyle {\alpha ,k,\zeta >0}\) are given real numbers.
The initial condition is given by
We suppose that f is a locally \(L_1\)-Lipschitz function and there exists a positive constant \(\sigma \), such that
Remark 4.1
Under the condition (4.3) and for \(\nabla v=0\), we get
We introduce the notion of capacity solution as follows:
Definition 4.2
The pair (u, f) is called a capacity solution for the problem (1.1), if the following conditions hold:
-
(1)
\(u\in \textit{F}\) and \(\nabla u\in L_{\bar{\psi }}(\Omega )^N\).
-
(2)
(u, f) satisfies the following equation:
$$\begin{aligned} \displaystyle {\dfrac{\partial u}{\partial s}-\Delta u= \dfrac{\lambda f(u)}{(\int _{\Omega }f(u)\mathrm{{d}}x)^{2}}, \text{ in } Q_{S}.} \end{aligned}$$ -
(3)
\(u(.,0)=u_0,\) in \(\Omega \).
We obtain using the Lemma 3.5 and the regularity of u that \(u\in C(]0,S [ ;L^1(\Omega ))\). Then, u is well defined in \(L^1(\Omega )\).
5 An existence result
In this section, we develop the proof of the main result.
Theorem 5.1
We assume that hypotheses (2.7), (2.8), (2.11), (2.12) and (4.2)–(4.4) hold, then the problem (1.1) has a capacity solution in the sense of Definition 4.2.
To prove this result, we need to apply the following theorem.
Theorem 5.2
We suppose that the conditions (2.7), (2.8), (4.3) and (4.4) hold. Then, there exists a weak solution for the problem (1.1), that is
for each \(\displaystyle {\varphi \in W^{1, x}_0 L_{\psi }\left( Q_S\right) } \) and \(s\in [ 0,S]\).
Proof
To show the existence of a weak solution, Schauder’s fixed point theorem will be applied. To this end, applying Theorem 3.7, we get the existence of a solution to the following problem, for all \(v\in W^{1, x}_0 L_{\psi }\left( Q_S\right) :\)
From (4.5), we obtain
We have to prove that
where K is a positive constant.
In view of the fact that \(L^2(\Omega )\hookrightarrow H^{-1}(\Omega )\), we get
Since f is a Lipshitz function, then we get
All terms in the right-hand side are bounded due to \(w\in E_{\psi }\left( Q_S\right) \hookrightarrow L^1(0,S;H^{-1}(\Omega )).\) Then, there exists a positive constant C, such that
Hence
Using the following continuous embedding \(\displaystyle { L^2\left( 0,S;H^{-1}(\Omega )\right) \hookrightarrow W^{-1, x} E_{\bar{\psi }}\left( Q_S\right) }\) obtained by applying Lemma 2.12 and Remark 2.13, we get that:
Now, we are in a position to employ Theorem 3.7, and we get the existence of a weak solution. Now, we prove that \(\displaystyle {\mid \nabla u\mid \in F_{\psi }(\Omega )}\) and the following estimates:
where C and \(C_2\) are a positive constant. Let us prove (5.4). To this end, we use (4.3), to obtain
Hence
From (5.1), we get
Using (4.5) and the hypothesis on f, we get that
Owing to \(\displaystyle {L_\psi (\Omega )\hookrightarrow L^2(\Omega )}\) and \(w\in E_{\psi }\left( Q_S\right) \hookrightarrow L^{1}\left( 0, S ; E_{\psi }(\Omega )\right) \hookrightarrow L^1(0,S;L^2(\Omega )),\) there exists a positive constant C, such that
From (5.7) and (5.9), we obtain
it yields that \(\mid \nabla u\mid \in F_{\psi }(\Omega ).\)
Now, we state to prove the inequality (5.5). Knowing that
This implies that
Applying (5.9) and using (4.1), we get
for each \(\varphi \in W_{0}^{1, x} E_{\psi }\left( Q_S\right) \) where \(\Vert \nabla \varphi \Vert _{\psi ,Q_S}=\dfrac{1}{k+1}\). Consequently, we get
from whence follows, there exists a positive constant \(C_2\), such that
It yields that \(\Vert \nabla u\Vert _{\bar{\psi },Q_S}\le ~ C_2,\) as a consequence \(\nabla u\in E_{\bar{\psi }}(Q_S);\) hence, \(\Delta u\in W_{0}^{-1, x}E_{\bar{\psi }}(Q_S).\) Keeping this in mind, using the following inclusion:
and the first equation of the problem (5.1), it follows that:
We introduce the following operator:
where u is the solution for the problem (5.1). G is compact operator. Indeed, \(\textit{F}\subset E_{\varphi }\left( Q_S\right) \) (i.e., \( \textit{F}\) is included in \(E_{\varphi }\left( Q_S\right) \) with the compact injection), we can show this embedding using Lemmas 3.6 and 3.4. From the inequality (5.11), we find that set \(\displaystyle {\left\{ \frac{\partial u}{\partial s} ;~~ u\in \textit{F} \right\} }\) is bounded in \(W^{-1, x} L_{\bar{\psi }}\left( Q_S\right) \). From Lemmas 3.6 and 3.4, where \(Y:=L^1(\Omega )\), we get the following compact embedding \(\textit{F}\hookrightarrow E_{\varphi }\left( Q_S\right) \). This combined with (5.11) and (5.10) yields to the compactness of the mapping G.
We define
\(B_\nu \) is bounded and closed. Keeping this and (5.10) in mind, we obtain \(G(B_\nu )\subset B_\nu \).
To achieve the proof of the existence of a weak solution, it suffices to show that G is a continuous operator. To this end, we assume that \((v_n)_{n\in \mathbb {N}}\subset B_\nu \), such that \(v_n\longrightarrow \omega \), also, let us consider \(G(\omega )=u\) and \(G(v_n)=u_n\). Hence
Since \( L^2(Q_S)\) is a Banach space, then \(v_n\longrightarrow \omega \) in \( L^2(Q_S),\) so there exists a subsequence still denoted by \((v_n)_{n\in \mathbb {N}}\), such that \(v_n\longrightarrow \omega \) a.e in \(Q_S\). Knowing that
Then, \((v_n)_{n\in \mathbb {N}}\) is bounded. Hence, there exists a subsequence, such that
and
We choose \(v=u_n-u\) in (5.1), and we obtain
By subtracting the above two equations, we get
On the other hand, we have the following identity:
Then, from (5.15) and (5.16), it yields that
Putting
Then
Hence
Then
Knowing that \((u_n)_{n\in \mathbb {N}}\) is bounded in \(L^2(\Omega )\), by applying the convergence dominate theorem, we get
Combining (5.17) with (5.18), we get
From (5.17)–(5.20), we get \( u_n\xrightarrow {\parallel .\parallel _{L^2(\Omega )}} u.\) Knowing that \( u_n\longrightarrow V \text{ in } E_{\varphi }(Q_S)\subset L_{\varphi }(Q_S) \subset L^2(Q_S)\), we obtain that \( u_n\xrightarrow {\parallel .\parallel _{L^2(\Omega )}} V.\) This implies that \(V=u,\) then \(G(v_n) \longrightarrow G(\omega )=u\). Hence, G is continuous. This completes the proof of Theorem 5.2. \(\square \)
We now proceed to prove Theorem 5.1.
Proof
The proof consists of four steps. We begin by presenting a sequence of approximation problems, establishing a priori estimates for them, and demonstrating intermediate results, namely strong convergence in \(L^1(\Omega )\) of \((\nabla u_n)_{n\in \mathbb {N}}.\)
Step I
For every \(n\in \mathbb {N}\), we consider the following approximate problem:
Under the assumption (4.1), we obtain
From assumptions (4.2)–(4.3), we get
Applying Theorem 5.2, to get the existence of a weak solution to the approximate problem (5.21). We use \(u_n\) as a test function in (5.21). Then, we get
Hence
Consequently
Keeping this and (5.6) in mind, we obtain
On the other hand, using the condition (4.5) and H\(\ddot{o}\)lder’s inequality, we get
Since \((u_n)_{n\in \mathbb {N}}\) is a bounded sequence in \(L^2(Q_S)\). Then, there exists a positive constant \(C_5\), such that
Recall from Remark 2.5 that
this implies that \((u_n)_{n\in \mathbb {N}}\) is a bounded sequence in \(W^{1, x}_0 L_{\psi }\left( Q_S\right) \). Then, there exists a subsequence still denoted \((u_n)_{n\in \mathbb {N}}\) weakly converges in \(W^{1, x}_0 L_{\psi }\left( Q_S\right) \) as \(n\longrightarrow \infty \) to a limit u, such that
On the other hand, for any function \(\varphi \in W^{1, x}_0 E_{\psi }\left( Q_S\right) ^N\), such that \(\parallel \nabla \varphi \parallel _{\psi ,Q_S}=\dfrac{1}{m+1}\) where m is a positive real number, we have
Using the equivalence between the Luxemburg norm and the Orlicz norm, and using (5.22), there exists a positive constant \(C_6\), such that
from whence follows, \((\nabla u_n)_{n\in \mathbb {N}}\) is bounded in \(L_{\bar{\psi }}\left( Q_S\right) ^N\). This implies that there exists a subsequence still denoted \(( \nabla u_n)_{n\in \mathbb {N}}\), such that
Since the sequences \((\nabla u_n)_{n\in \mathbb {N}}\) and \(\left( \lambda \dfrac{f(u_n)}{(\int _{\Omega }f(u_n)\mathrm{{d}}x)^{2}} \right) _{n\in \mathbb {N}}\) are bounded in \( W^{-1, x} L_{\bar{\psi }}\left( Q_S\right) \). Hence, using the first equation of the problem (5.21), we get that the sequence \(\left( \dfrac{\partial u_n}{\partial s}\right) _{n\in \mathbb {N}} \) is bounded in \( W^{-1, x} L_{\bar{\psi }}\left( Q_S\right) \). Consequently, \(( u_n)_{n\in \mathbb {N}} \) is bounded in \(\textit{F}\). In view of the fact that \(\textit{F}\hookrightarrow E_\phi \left( Q_S\right) \) is compact, then, for a subsequence still denoted in the same way, we get
where \(u\in W^{1, x}_0 L_{\psi }\left( Q_S\right) \) is also the same limit appearing in (5.23).
Step II
We introduce the following regularized sequences for \(i,j\in \mathbb {N}\):
-
(1)
\(v_{j} \rightarrow u\) in \(W_{0}^{1, x} L_{\psi }\left( Q_S\right) \) with the modular convergence;
-
(2)
\(v_{j} \rightarrow u\) and \(\nabla v_{j} \rightarrow \nabla u\) a.e in \(Q_{S}\);
-
(3)
\(\omega _i \rightarrow u_{0}\) in \(L^{2}(\Omega )\) with the strong convergence;
-
(4)
\(\left\| \omega _{i}\right\| _{L^{2}(\Omega )} \le 2\left\| u_{0}\right\| _{L^{2}(\Omega )},\) for all \(i \ge 1\).
These fourth points are satisfied for all \(\omega \in D(\Omega )\) and \(v_{j}\in D(Q_{S})\).
Let \(R>0\) a real number, and we define the truncation function as in (2.15). Then, for each \(R,\eta >0\) and for \(i,j\in \mathbb {N}\), we consider the function \(\omega ^i_{\eta ,j}\in W^{1, x}_0 L_{\psi }\left( Q_S\right) \) defined as follows \(\omega ^i_{\eta ,j}:=S_{R}(v_j)_\eta +\exp (-\eta s)S_{R}(\omega _j)\), such that \(S_{R}(v_j)_\eta \) is the mollification with respect to the time variable of \(S_{R}(v_j)\) appearing in (3.1). From Lemma 3.1, it follows that:
with the modular convergence in the two last convergences. \(\square \)
We consider subsequences in (5.26)–(5.28), without loss the generality that convergences in (5.26)–(5.28) hold a.e. in \( Q_S\).
Proposition 5.3
Let \(u_n\) be a solution to the problem (5.21). Then, for a subsequence, we have the following convergence:
Proof
Throughout this paper, we use \(\chi _{r}^j\) and \(\chi _{r}\) as the characteristic functions of the following sets:
For any real numbers \(\eta ,\vartheta >0\) and for \(i,j,n\in \mathbb {N}\), we use the admissible test function \(\phi _{n,j,\vartheta }^{\eta ,i}:=S_{\vartheta }(u_n-\omega ^i_{\eta ,j})\) in the first equation of the approximate problem (5.21), and we get
On the other hand, using the condition (4.5), and by the same reasoning done to get (5.2), we obtain
From (2.15), we obtain
Using (5.31), we get
Now, we split the first term on the left side of the above inequality into two parts and estimate each one separately
We start by estimating the first term on the right side of the above identity
From (2.16), it can be shown that
It follows that:
Then, for all \(\eta ,\vartheta >0\) and \(i,j,n\ge 1\), and from (5.34), we get
Now, we derive an estimate for the second term on the right side of (5.33). Under assumption (5.26), we get
Then
Under hypotheses (5.26)–(5.27), we have \(\vert \omega ^i_{\eta }\vert \le R\), and due to \(rS_{\vartheta }(r)\ge 0,~r\in \mathbb {R}\), we get that for all \(\eta ,\vartheta ,R>0\) and \(i\ge 1\)
Keeping this and (5.35) in mind, we get
On the other hand, we have
Then
where
Under assumption (4.3), we have
Then, \(\displaystyle {I_{i,j,n,\vartheta }\ge I_{0,i,j,n,\vartheta }-I_{2,i,j,n,\vartheta }}\); from (5.26), we have \(\mid \omega ^i_{\eta ,j}\mid \le R, \text{ a.e. } \text{ in } ~Q_{S},\) and this implies that
It follows that for \(n>R+\vartheta \):
which gives
Using again inequality (5.39), we get by definition \(S_{\vartheta +R}(u_n)=u_n\), and hence, \(\nabla S_{\vartheta +R}(u_n)=\nabla u_n\). In view of the fact that \((\nabla u_n)_{n\in \mathbb {N}}\) is bounded in \(L_{{\psi }}(Q_S)^N\), we get that \((\nabla S_{\vartheta +R}(u_n))_{n\in \mathbb {N}}\) is also bounded in \(L_{{\psi }}(Q_S)^N\). Then, there exists \(a_1\), such that \( \nabla S_{\vartheta +R}(u_n)\rightharpoonup a_1\) as \(n\longrightarrow \infty \) in \(L_{{\psi }}(Q_S)^N\) for the weak topology \(\displaystyle {\sigma (\prod L_{{\psi }},\prod E_{\bar{\psi }} )} \). In view of the fact that
strongly in \(E_\psi (Q_S)^N\) as \(n\longrightarrow \infty \), we get
Under assumptions (5.27) and (5.28), we obtain
as \(n,j\longrightarrow \infty \). We apply Lemma 2.8, and letting \(\vartheta ,j\longrightarrow \infty \), we obtain
For \(\vert u\vert >R\), we get \(S_{R}(u)=0,\) which yields \(I_3=0\). Thus
By using (5.40), we obtain
Using (5.32), we get
Owing to (5.38), it follows that:
Then
We have
where \(A:=\lbrace \vert S_{R}(u_n)-\omega ^i_{\eta ,j}\vert \le \vartheta \rbrace .\)
Now, we show that \(I_{5,i,j,n,\eta }\longrightarrow 0\). Knowing that \((S_{R}(u_n))_{n\in \mathbb {N}}\) is bounded. Hence, there exits \(a_0\), such that
Since
It follows that \((\varGamma ^n(v_j))_{n\in \mathbb {N}}\) is bounded in \(E_{\bar{\psi }}(Q_S)\). Hence, \((\nabla S_{R}(u_n)\varGamma ^n(v_j))_{n\in \mathbb {N}}\) is bounded as well. Applying the dominated convergence theorem, we get
Recall (5.40) and (5.42), we get
From (5.41) and (5.43), we can take \( \epsilon (n, i, \eta ,j ) :=I_{2,i,j,n,\vartheta } -I_{5,i,j,n,\eta }\) where \(\epsilon (n, i, \eta ,j )\longrightarrow 0 \text{ as } i, j,n,\eta \longrightarrow \infty \). This implies that \(I_{4,i,j,n,\eta }\le C\vartheta +\epsilon (n, i, \eta ,j )\).
Putting
which is a nonnegative quantity. Since \((\nabla S_{R}(u_n))\) is bounded in \(L_{{\psi }}(Q_S)^N\), then the same holds for \(N_n\). Let us \(J_n^r:=\int _{Q_r}N_n^\theta \textrm{ d} x \textrm{ d} s\) for each \(\theta \) in ]0, 1[, we get
Using H\(\ddot{o}\)lder’s inequality, we obtain
It follows that:
On the other hand, for \(s\ge r\) and \(r>0\), we have
where
Then
Putting
By virtue of \(\nabla S_{R}(v_j) \chi _{j}^s\longrightarrow \nabla S_{R}(u) \chi ^s\) as \(j\longrightarrow \infty \) and \(\nabla S_{R}(u_n)\rightharpoonup \nabla S_{R}(u)\) weakly in \(E_\psi (Q_S)^N\), it follows that \(M_{j,n}\longrightarrow 0\) as \(n,j\longrightarrow \infty \). Since \((\nabla S_{R}(v_j) \chi _{j}^s)_j\) converges to \(\nabla S_{R}(u) \chi _s\) strongly in \(E_\psi (Q_S)^N\). Applying the dominated convergence theorem, we get \(J_{4,n,j}\longrightarrow 0\) as \(n,j\longrightarrow \infty \). For \(J_{3,n,j}\), knowing that sequence \((\nabla S_{R}(u_n))_{n\in \mathbb {N}}\) is bounded and \((\nabla S_{R}(v_j) \chi _{j}^s)_j\) converge strongly to \(\nabla S_{R}(u) \chi ^s\) in \(E_\psi (Q_S)^N\), then \(\left( \nabla S_{R}(u_n)-\nabla S_{R}(v_j)\chi _{j}^s\right) \) is bounded. Using again the convergence of \((\nabla S_{R}(v_j) \chi _{j}^s)_j\) to \(\nabla S_{R}(u) \chi ^s\), then \(J_{3,n,j}\longrightarrow 0\) as \(n,j\longrightarrow \infty \). For \(J_{2,n,j}\), we have \(\left( \nabla S_{R}(v_j)\chi _{j}^s-\nabla S_{R}(u)\chi _{s}\right) \longrightarrow 0\) as \(j\longrightarrow \infty \), \((\nabla S_{R}(u_n))_{n\in \mathbb {N}}\) and \((\nabla S_{R}(v_j) \chi _{j}^s)_j\) are convergent. Consequently, \(\left( \nabla S_{R}(u_n)-\nabla S_{R}(v_j) \chi _{j}^s\right) ,\) is bounded. Applying again the convergence dominate theorem, we get \(J_{2,n,j}\longrightarrow 0\) as \(n,j\longrightarrow \infty \), and
Letting n, j, then \(\eta ,i,s,\vartheta \) to infinity, we get
On the other hand, from (4.3), we obtain
We recall that
almost everywhere in \(Q_r\). Since \(r>0\) is arbitrary, we recall that for another subsequence \(\nabla S_{R}(u_n)\longrightarrow \nabla S_{R}(u)\) almost everywhere in \(Q_S\). Finally, for \(R>0\) arbitrary, we get
This concludes the proof. \(\square \)
Step III
We obtain the first condition of Definition 4.2, by applying (5.31), (5.24), and (5.26). For the second condition of the same definition obtained using the convergence (5.44) and the smoothness of the function f. For the regularity of the solution u, we use (5.11) and we apply directly Lemma 3.5, we get that \(u\in C\left( [0,S]; L^1(\Omega )\right) \). This concludes the proof of Theorem 5.1\(\square \) .
6 Conclusion and perspectives
In this paper, we showed the existence result for a capacity solution to a nonlocal thermistor problem in Musielak–Orlicz–Sobolev spaces. In the future, we plan on studying the regularity of a global attractor. Other intriguing problems about this capacity solution surround the development of specific qualitative properties [5], such as the calculation of an energy estimate, the study of long-term behavior, or even the possibility of a blow-up event.
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Dahi, I., Sidi Ammi, M.R. Existence of capacity solution for a nonlocal thermistor problem in Musielak–Orlicz–Sobolev spaces. Ann. Funct. Anal. 14, 12 (2023). https://doi.org/10.1007/s43034-022-00237-x
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DOI: https://doi.org/10.1007/s43034-022-00237-x
Keywords
- Existence
- Capacity solution
- Nonlinear parabolic equation
- Thermistor problem
- Musielak–Orlicz–Sobolev spaces