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Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term

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  • The authors of this paper study some singular phenomena (blowing-up or vanishing in finite time) of solutions to the initial homogeneous $\hbox{Neumann}$ problem of a nonlinear diffusion equation involving the $p(x,t)$-Laplace operator and a nonlinear source. The variable exponent $p(x,t)$ leads to the failure of some techniques, such as upper-lower solutions technique and the scaling method etc., in studying the problem; it also leads to the lack of some valuable properties such as the monotonicity of the energy integral etc. The authors construct a suitable control functional, improve the regularity of the approximate solutions and obtain a new energy inequality to prove that the solution of the problem with a positive initial energy blows up in finite time. Furthermore, under some appropriate conditions, the authors study the vanishing property and the extinction rate estimate of the solutions to the problem by establishing some inequalities the solutions satisfy. It is worth pointing out that the results are obtained with the assumption that $p_{t}(x,t)$ is only negative and integrable which is weaker than those the most of the other papers required.
    Mathematics Subject Classification: Primary: 35K55, 35K40; Secondary: 35B65.


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