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Critical global asymptotics in higher-order semilinear parabolic equations. (English) Zbl 1035.35048

Summary: We consider a higher-order semilinear parabolic equation \(u_t=-(-\Delta)^m u- g(x,u) \) in \(\mathbb{R}^N \times \mathbb{R}_+\), \( m \!>\! 1\). The nonlinear term is homogeneous: \(g(x,s u) \equiv | s| ^{P-1}s g(x,u)\) and \(g(s x,u) \equiv | s| ^{Q} g(x,u)\) for any \(s \in \mathbb{R}\), with exponents \(P>1\), and \(Q >-2m\). We also assume that \(g\) satisfies necessary coercivity and monotonicity conditions for global existence of solutions with sufficiently small initial data. The equation is invariant under a group of scaling transformations. We show that there exists a critical exponent \(P=1+(2m+Q)/N\) such that the asymptotic behavior as \(t \rightarrow\infty\) of a class of global small solutions is not group-invariant and is given by a logarithmic perturbation of the fundamental solution \(b(x,t)=t^{-N/2m}f(xt^{-1/2m})\) of the parabolic operator \(\partial/\partial t +(-\Delta)^m\), so that for \(t \!\gg\! 1\), \(u(x,t) \!=\! C_0 (\ln t)^{-N/(2m+Q)}[b(x,t) + o(1)]\), where \(C_0\) is a constant depending on \(m\), \(N\), and \(Q\) only.

MSC:

35K30 Initial value problems for higher-order parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35K65 Degenerate parabolic equations
35B33 Critical exponents in context of PDEs
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