Boundedness of global solutions of nonlinear diffusion equations. (English) Zbl 0764.35010

Fila, Marek

J. Differ. Equations 98, No. 2, 226-240 (1992).

This paper deals with large time behaviour of solutions of Dirichlet- initial value problems for the equation \(u_ t=\Delta(| u|^{m- 1}u)+f(u)\) when \(f\) grows superlinearly. In particular the question of global existence without uniform a priori bounds is examined. The case when \(m=1\) and \(f\) is allowed to grow exponentially is of particular concern and it is shown that global existence without uniform bounds is impossible in one and two dimensions.

Reviewer: J.F.Toland (Bath)

MSC:

35B35	Stability in context of PDEs
35K15	Initial value problems for second-order parabolic equations
35K55	Nonlinear parabolic equations
35K05	Heat equation

Keywords:

blow-up; large time behaviour; Dirichlet-initial value problems; global existence

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