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Absence of solutions to higher-order evolution differential inequalities. (Russian, English) Zbl 1050.35160

Sib. Mat. Zh. 44, No. 1, 143-159 (2003); translation in Sib. Math. J. 44, No. 1, 117-131 (2003).
Solvability is studied for higher-order evolution differential inequalities \[ \begin{gathered} \frac{\partial^ku}{\partial t^k}-\Delta u\geq | x| ^\sigma | u| ^q\quad \text{in } \Omega\times(0,\infty), \\ u| _{\partial B_R\times(0,\infty)}\geq0, \quad \frac{\partial^{k-1}u}{\partial t^{k-1}}\biggr| _{t=0}\geq0 \end{gathered} \] or \[ \begin{gathered} \frac{\partial^ku}{\partial t^k}-\Delta u\geq t^\gamma | u| ^q \quad \text{in } \Omega\times(T,\infty), \\ u| _{\partial B_R\times(0,\infty)}\geq0, \quad \frac{\partial^{k-1}u}{\partial t^{k-1}}\biggr| _{t=T}\geq0, \end{gathered} \] \(\Omega={\mathbb R}^N\setminus B_R\), \(B_R=\{x\in{\mathbb R}^N:| x| \leq R\}\), \(R>0\), \(k\geq1\), \(T>0\), \(\sigma>-2\), \(-k<\gamma\leq0\). Conditions on the exponent  \(q\) are stated under which these inequalities have no nontrivial global weak solutions. In the corresponding particular cases these conditions coincide with those of Fujita and Hayakawa for the parabolic case and those of Kato for the hyperbolic case.

MSC:

35R45 Partial differential inequalities and systems of partial differential inequalities
35G30 Boundary value problems for nonlinear higher-order PDEs
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