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Convergence and sharp thresholds for propagation in nonlinear diffusion problems. (English) Zbl 1207.35061
Authors’ abstract: We study the Cauchy problem
$u_t= u_{xx}+ f(u), \qquad u(0, x) = u_{0}(x),$
where $$f(u)$$ is a locally Lipschitz continuous function satisfying $$f(0) = 0$$. We show that any nonnegative bounded solution with compactly supported initial data converges to a stationary solution as $$t\rightarrow \infty$$. Moreover, the limit is either a constant or a symmetrically decreasing stationary solution. We also consider the special case where $$f$$ is a bistable nonlinearity and the case where $$f$$ is a combustion type nonlinearity. Examining the behavior of a parameter-dependent solution $$u_\lambda$$, we show the existence of a sharp threshold between extinction (i.e., convergence to 0) and propagation (i.e., convergence to 1). The result holds even if $$f$$ has a jumping discontinuity at $$u=1$$.

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35K15 Initial value problems for second-order parabolic equations 35K58 Semilinear parabolic equations 80A25 Combustion
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