We investigate in this paper propagation phenomena for the heterogeneous reaction-diffusion equation∂ t u - Δ u = f (t, u), x ∈ R N, t ∈ R, where f = f (t, u) is a KPP monostable nonlinearity which depends in a general way on t ∈ R. A typical f which satisfies our hypotheses is f (t, u) = μ (t) u (1 - u), with μ ∈ L ∞ (R) such that ess inf t ∈ R μ (t) > 0. We first prove the existence of generalized transition waves (recently defined in Berestycki and Hamel (2007) ) for a given class of speeds. As an application of this result, we obtain the existence of random transition waves when f is a random stationary ergodic function with respect to t ∈ R. Lastly, we prove some spreading properties for the solution of the Cauchy problem.
Propagation phenomena for time heterogeneous KPP reaction–diffusion equations
ROSSI, LUCA
2012
Abstract
We investigate in this paper propagation phenomena for the heterogeneous reaction-diffusion equation∂ t u - Δ u = f (t, u), x ∈ R N, t ∈ R, where f = f (t, u) is a KPP monostable nonlinearity which depends in a general way on t ∈ R. A typical f which satisfies our hypotheses is f (t, u) = μ (t) u (1 - u), with μ ∈ L ∞ (R) such that ess inf t ∈ R μ (t) > 0. We first prove the existence of generalized transition waves (recently defined in Berestycki and Hamel (2007) ) for a given class of speeds. As an application of this result, we obtain the existence of random transition waves when f is a random stationary ergodic function with respect to t ∈ R. Lastly, we prove some spreading properties for the solution of the Cauchy problem.Pubblicazioni consigliate
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