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Blow up for some nonlinear parabolic problems with convection under dynamical boundary conditions

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  • Blow up phenomena for solutions of nonlinear parabolic problems including a convection term $\partial_tu = div(a(x)\Deltau) + f(x, t, u,\Deltau)$ in a bounded domain under dissipative dynamical time lateral boundary conditions $\sigma\partial_tu + \partial_vu = 0$ are investigated. It turns out that under natural assumptions in the proper superlinear case no global weak solutions can exist. Moreover, for certain classes of nonlinearities the blow up times can be estimated from above as in the reaction diffusion case [6]. Finally, as a model case including a sign change of the convection term, the occurrence of blow up is investigated for the one–dimensional equation $\partial_tu = \partial^2_xu − u\partial_xu + u^p.
    Mathematics Subject Classification: 35B05, 35B40,35K55, 35K57, 35K65, 35R45.


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