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A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions. (English) Zbl 1066.35045

Summary: We study the localization properties of weak solutions to the Dirichlet problem for the degenerate parabolic equation \[ u_t-\text{div} \bigl(|u|^{\gamma(x,t)}\nabla u\bigr)=f, \] with variable exponent of nonlinearity \(\gamma\). We prove the existence and uniqueness of weak solutions and establish conditions on the problem data and the exponent \(\gamma(x,t)\) sufficient for the existence of such properties as finite speed of propagation of disturbances, the waiting time effect, finite time vanishing of the solution. It is shown that the solution may instinct in a finite time even if \(\gamma\equiv \gamma (x)\geq 0\) in the problem domain but \(\max\gamma=0\).

MSC:

35K65 Degenerate parabolic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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