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On blow-up results for solutions of inhomogeneous evolution equations and inequalities. II. (English) Zbl 1212.35522

Summary: The main purpose of this work is to further develop ideas and methods from a recent paper by the authors [J. Math. Anal. Appl. 290, No. 1, 76–85 (2004; Zbl 1061.35181)]. In particular, we obtain new blow-up results for solutions of the inequality \[ | u| _t \geq \Delta [| u| ^{\sigma }u] + | u | ^q + \omega (x) \] on the half-space \(\mathbb R^1_+ \times \mathbb R^n\), where \(n\geq 1\), \(\sigma \geq 0\), \(q>1+\sigma \), and \(\omega \:\mathbb R^n\to \mathbb R^1\) is a globally integrable function such that \(\int _{\mathbb R^n}\omega (x)\,\text dx>0\), and establish that for \(n>2\) the critical blow-up exponent \(q^* = n(1+\sigma)/(n-2)\) is of the blow-up type.

MSC:

35R45 Partial differential inequalities and systems of partial differential inequalities
35B44 Blow-up in context of PDEs
35K55 Nonlinear parabolic equations
31C45 Other generalizations (nonlinear potential theory, etc.)
58J35 Heat and other parabolic equation methods for PDEs on manifolds

Citations:

Zbl 1061.35181
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