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Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part I: A general quasiconvergence theorem and its consequences. (English) Zbl 1345.35052
Summary: We consider the Cauchy problem \[ u_1 = u_{xx} + f(u), \;x\in \mathbb R, t>0, \]
\[ u(x,0)=u_0 (x), \;x \in \mathbb R, \] where \(f\) is a locally Lipschitz function on \(\mathbb R \) with \(f(0)=0\), and \(u_0\) is a nonnegative function in \(C_0(\mathbb{R})\), the space of continuous functions with limits at \(\pm\) equal to \(0\). Assuming that the solution \( u\) is bounded, we study its large-time behavior from several points of view. One of our main results is a general quasiconvergence theorem saying that all limit profiles of \( u(\cdot,t)\) in \( L^{\infty}_{loc} (\mathbb R) \) are steady states. We also prove convergence results under additional conditions on \( u_0\). In the bistable case, we characterize the solutions on the threshold between decay to zero and propagation to a positive steady state and show that the threshold is sharp for each increasing family of initial data in \( C_0(\mathbb R)\).

MSC:
35K15 Initial value problems for second-order parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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