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

Xin, Jack

Methods Appl. Anal. 10, No. 2, 191-197 (2003).

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)

Keywords:

Kolmogorov-Petrovsky-Piskunov nonlinearity; almost sure Lyapunov exponents; homogenized Hamiltonians