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Variable exponent perturbation of a parabolic equation with \(p(x)\)-Laplacian. (English) Zbl 1438.35255
Summary: This paper is concerned with the study of the global existence and the decay of solutions of an evolution problem driven by an anisotropic operator and a nonlinear perturbation, both of them having a variable exponent. Because the nonlinear perturbation leads to difficulties in obtaining a priori estimates in the energy method, we had to significantly modify the Tartar method. As a result, we could prove the existence of global solutions at least for small initial data. The decay of the energy is derived by using a differential inequality and applying a non-standard approach.

MSC:
35K92 Quasilinear parabolic equations with \(p\)-Laplacian
35K58 Semilinear parabolic equations
35K55 Nonlinear parabolic equations
35L60 First-order nonlinear hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations
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