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Blow-up of solutions of a semilinear heat equation with a visco-elastic term. (English) Zbl 1088.35030
Chipot, Michel (ed.) et al., Nonlinear elliptic and parabolic problems. A special tribute to the work of Herbert Amann, Zürich, Switzerland, June 28–30, 2004. Basel: Birkhäuser (ISBN 3-7643-7266-4/hbk). Progress in Nonlinear Differential Equations and their Applications 64, 351-356 (2005).

Summary: We consider an initial boundary value problem related to the equation

${u}_{t}-{\Delta }u+{\int }_{0}^{t}g\left(t-s\right){\Delta }u\left(x,s\right)ds={|u|}^{p-2}u$

and prove, under suitable conditions on $g$ and $p$, a blow-up result for solutions with negative or vanishing initial energy. This result improves an earlier one by the author.

##### MSC:
 35K65 Parabolic equations of degenerate type 35K60 Nonlinear initial value problems for linear parabolic equations 45K05 Integro-partial differential equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE)