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Critical exponents for the blowup of solutions with sign changes in a semilinear parabolic equation. II. (English) Zbl 0924.35055
Authors’ abstract: The blow-up of solutions of the Cauchy problem, $$\cases u_t= u_{xx}+ | u|^{p-1} u\quad &\text{in }\bbfR \times (0,\infty),\\ u(x,0)=u_0(x) \quad & \text{in }\bbfR, \endcases$$ is studied. Let $\Lambda_k$ be the set of functions on $\bbfR$ which change sign $k$ times. It is shown that for $$p_k=1+ {2\over k+1}, \quad k=0,1,2, \dots,$$ any solution with $u_0\in \Lambda_k$ blows up in finite time if $1<p\le p_k$, whereas a global solution with $u_0\in\Lambda_k$ exists if $p>p_k$. It is also shown that if $u_0$ decays more slowly than $| x|^{-2/(p-1)}$ as $| x | \to\infty$, then the solution blows up in finite time regardless of the number of sign changes. Part I, see Math. Ann. 307, No. 4, 663-675 (1997; Zbl 0872.35046).

35K15Second order parabolic equations, initial value problems
35B40Asymptotic behavior of solutions of PDE
35K57Reaction-diffusion equations
Full Text: DOI
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