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KPP front speeds in random shears and the parabolic Anderson problem. (English) Zbl 1052.35099

Summary: We study the asymptotics of front speeds of the reaction-diffusion equations with Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity and zero mean stationary ergodic Gaussian shear advection on the entire plane. By exploiting connections of KPP front speeds with the almost sure Lyapunov exponents of the parabolic Anderson problem, and with the homogenized Hamiltonians of Hamilton-Jacobi equations, we show that front speeds enhancement is quadratic in the small root mean square (RMS) amplitudes of white in time zero mean Gaussian shears, and it grows at the order of the large RMS amplitudes. However, front speeds diverge logarithmically if the shears are time independent zero mean stationary ergodic Gaussian fields.

MSC:

35K57 Reaction-diffusion equations
35R60 PDEs with randomness, stochastic partial differential equations
35B40 Asymptotic behavior of solutions to PDEs
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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