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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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