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Traveling fronts in space-time periodic media. (English) Zbl 1182.35074

The author deals with the existence of pulsating traveling fonts for the equation
\[ \partial _tu-\nabla \cdot (A(t,xt) \nabla u) +q(t,x)\cdot \nabla u=f(t,x,u). \]
The author supposes that the diffusion matrix \(A\), the advection term \(q\) and the reaction term \(f\) are periodic in \(t\) and \(x\) and proves that there exists a half-line of speeds \(\left[ c^{**},+\infty \right) \) associated with pulsating traveling fronts. Moreover, the author proves that there exists some speed \(c^{*}\) associated to some pulsating traveling front such that there exists no pulsating traveling front of speed \(c<c^{*}\), even if \(f \) is not KPP type. If \(f\) satisfies a KPP type assumption, then \(c^{*}=c^{**} \) and the author characterizes the minimal speed \(c^{*}\).
In addition the author proves that if \(s\mapsto f\left( t,x,s\right) /s\) is nonincreasing, then there exist some Lipschitz continuous pulsating traveling fronts of speed \(c\) if and only if \(c\geq c^{*}.\)

MSC:

35C07 Traveling wave solutions
35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
35B50 Maximum principles in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35P15 Estimates of eigenvalues in context of PDEs
35B10 Periodic solutions to PDEs
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