Blow Up in a Semilinear Pseudo-Parabolic Equation with Variable Exponents
Annali dell'Università di Ferrara Scienze Matematiche/Annali dell'Università di Ferrara Sezione 7 Scienze matematiche(2019)
Abstract
In this paper, we consider the following pseudo-parabolic equation with variable exponents: $$\begin{aligned} u_{t}-\Delta u-\Delta u_{t}+\int _{0}^{t}g(t-\tau )\Delta u(x,\tau )d \tau = |u|^{p(x)-2}u. \end{aligned}$$ Under suitable assumptions on the initial datium $$u_{0}$$ , the relaxation function g and the variable exponents p, we prove that any weak solution, with initial data at arbitrary energy level, blows up in finite time.
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Key words
Pseudo-parabolic equation,Blow up,Finite time,Weak solutions,Variable-exponents nonlinearity,35B44,35D30,35L70
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