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Vanishing solutions of anisotropic parabolic equations with variable nonlinearity. (English) Zbl 1183.35177

Summary: We study the property of finite time vanishing of solutions of the homogeneous Dirichlet problem for the anisotropic parabolic equations
\[ u_t- \sum_{i=1}^n D_i \big(a_i(x,t,u)|D_iu|^{p_i(x,t)-2}D_iu\big)+ c(x,t)|u|^{\sigma(x,t)-2}u= f(x,t), \]
with variable exponents of nonlinearity \(p_i(x,t), \sigma(x,t)\in(1,\infty)\). We show that the solutions of this problem may vanish in a finite time even if the equation combines the directions of slow and fast diffusion and estimate the extinction moment in terms of the data. If the solution does not identically vanish in a finite time, we estimate the rate of vanishing of the solution as \(t\to\infty\). We establish conditions on the nonlinearity exponents which guarantee vanishing of the solution at a finite instant even if the equation eventually transforms into the linear one.

MSC:

35K65 Degenerate parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K59 Quasilinear parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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