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On the necessary conditions of global existence to a quasilinear inequality in the half-space. (English. Abridged French version) Zbl 0943.35110
Summary: We consider functions \(u(t,x)\) satisfying the inequality: \[ {\partial u\over\partial t}\geq L[u]^p+|u|^q,\quad p>0,\;q>1,\;q>p, \] for all \(t\geq 0\), \(x\in\mathbb{R}^n\), where \(L[v]:= \sum_{|\alpha|= m}D^\alpha(a_\alpha(t, x,v)v)\) is a differential operator of order \(m\), \(a_\alpha(t, x,v)\in L^\infty\). We prove that \(u(t,x)\equiv 0\) if \(1< q\), \(0< p< q\leq p+m/n\), \(u\in L^q_{\text{loc}}\), \(\int_{\mathbb{R}^n} u(0,x) dx\geq 0\).
This result generalizes some results of H. Fujita, K. Hayakawa, V. A. Galaktionov, S. P. Kurdyumov, A. P. Mikhailov, A. A. Samarskii, V. A. Kondratiev and S. D. Eidelman. We prove also a similar result for systems of inequalities. Another theorem generalizes a theorem of P. Meier.

MSC:
35R45 Partial differential inequalities and systems of partial differential inequalities
35G20 Nonlinear higher-order PDEs
35K55 Nonlinear parabolic equations
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