Stochastic Burgers-Huxley Equations: Global Solvability, Large Deviations and Ergodicity

Abstract: In this work, we consider the stochastic Burgers-Huxley equation perturbed by multiplicative Gaussian noise, and discuss about the global solvability results and asymptotic behavior of solutions. We show the existence of a global strong solution of the stochastic Burgers-Huxley equation, by making use of a local monotonicity property of the linear and nonlinear operators and a stochastic generalization of localized version of the Minty-Browder technique. We then discuss about the inviscid limit of the stochastic Burgers-Huxley equation to Burgers as well as Huxley equations. By considering the noise to be additive Gaussian, Exponential estimates for exit from a ball of radius $R$ by time $T$ for solutions of the stochastic Burgers-Huxley equation is derived, and then studied in the context of Freidlin-Wentzell type large deviations principle. Finally, we establish the existence of a unique ergodic and strongly mixing invariant measure for the stochastic Burgers-Huxley equation with additive Gaussian noise, using the exponential stability of solutions.