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Existence results of renormalized solutions for nonlinear \(p(\cdot )\)-parabolic equations with possibly singular measure data

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Abstract

We study the existence of renormalized solutions to a nonlinear parabolic boundary value problem with a general and possibly singular measure data, whose model is

$$\begin{aligned} ({\mathcal {P}})\ {\left\{ \begin{array}{ll} \frac{\partial b(u)}{\partial t} -\Delta _{p(x)}u =\mu &{}\ \ \text{ in }\ Q:=\Omega \times (0,T), \\ b(u)(t=0)=b(u_{0})(x) &{}\ \ \text{ in }\ \Omega ,\\ u(x,t)=0 &{}\ \ \text{ on }\ \partial \Omega \times (0,T),\\ \end{array}\right. } \end{aligned}$$

where \(\Omega\) is an open bounded subset of \({\mathbb {R}}^{N}\) (\(N\ge 2\)), \(T>0\), b is an increasing \(C^{1}\)-function, \(\Delta _{p(x)}u:=\text {div}(|\nabla u|^{p(x)-2}\nabla u)\) (\(1<p_{-}\le p(x)\le p_{+}<N\)) is the p(x)-Laplacian operator which, roughly speaking, behaves as \(|\nabla u|^{p(x)-1}\), \(\mu\) is a bounded Radon measure with bounded total variation on Q and \(b(u_{0})\) is an integrable function. We provide the assumptions, the notions of solution we are adopting and the statements of the existence result in the “generalized” Sobolev spaces with variable exponent using some “specific” decompositions on the data.

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Notes

  1. \(a(\cdot ,\cdot ,\zeta )\) is measurable on Q for every \(\zeta\) in \({\mathbb {R}}^{N}\), and \(a(x,t,\cdot )\) is continuous on \({\mathbb {R}}^{N}\) for a.e. (x, t) in Q.

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  1. LAMA, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, B.P. 1796, Atlas Fez, Morocco

    Khadija Moutaouakil, Jaouad Bennouna & Bouchra El Hamdaoui

  2. Faculté des Sciences Juridiques, Économiques et Sociales, Université Hassan 1, B.P. 764, Settat, Morocco

    Hicham Redwane

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  1. Khadija Moutaouakil
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Moutaouakil, K., Bennouna, J., El Hamdaoui, B. et al. Existence results of renormalized solutions for nonlinear \(p(\cdot )\)-parabolic equations with possibly singular measure data. Adv. Oper. Theory 6, 50 (2021). https://doi.org/10.1007/s43036-021-00139-0

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