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On the global solvability of semilinear parabolic systems with mixed right-hand side. (English. Russian original) Zbl 1074.35045
Function spaces, approximations, and differential equations. Collected papers dedicated to Oleg Vladimirovich Besov on his 70th birthday. Transl. from the Russian. Moscow: Maik Nauka/Interperiodika. Proceedings of the Steklov Institute of Mathematics 243, 59-79 (2003); translation from Tr. Mat. Inst. Steklova 243, 66-86 (2003).
The aim of the paper is to study the global solvability of the problem \[ u_t - L_1 u \geq b_1(t,x) u^P v^Q, \quad v_t - L_2 v \geq b_2(t,x) u^R v^S, \]
\[ ( u(t,x) \geq 0, \;v(t,x) \geq 0; \;x \in \mathbb R^N,\;t \geq 0) \]
\[ u(0,x) = u_0(x) \geq 0, \quad v(0,x) = v_0(x) \geq 0 \] with \(P,Q,R,S \geq 0\), where \(L_1\), \(L_2\) are second order elliptic differential operators in space variables with measurable coefficients and \(b_1, \; b_2\) are positive measurable functions. The solution is understood in the weak sense defined by means of an integral inequality. It is called degenerate if \(uv=0\) a.e. The author proves that the problem has no global nondegenerate nonnegative solutions if certain suppositions (too complicated to be formulated here) hold true.
The paper is organized as follows. Section 1 is devoted to the model problem \(L_1 = L_2 = \Delta\), \(b_1 = b_2 = 1\), in section 2 the general case is considered and in section 3 some examples are given. According to the author’s remark a complete list of references can be found in the survey [K. Deng and H. A. Levine, J. Math. Anal. Appl. 243, 85–126 (200; Zbl 0942.35025)] and in the monograph [E. Mitidieri and S. I. Pokhozhaev, A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova 234 (2001; Zbl 0987.35002)]
For the entire collection see [Zbl 1064.46002].
35K45 Initial value problems for second-order parabolic systems
35K55 Nonlinear parabolic equations
35R45 Partial differential inequalities and systems of partial differential inequalities