On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies

Abstract

In this work, we consider the forced generalized Burgers-Huxley equation and establish the existence and uniqueness of a global weak solution using a Faedo-Galerkin approximation method. Under smoothness assumptions on the initial data and external forcing, we also obtain further regularity results of weak solutions. Taking external forcing to be zero, a positivity result as well as a bound on the classical solution are also established. Furthermore, we examine the long-term behavior of solutions of the generalized Burgers-Huxley equations. We first establish the existence of absorbing balls in appropriate spaces for the semigroup associated with the solutions and then show the existence of a global attractor for the system. The inviscid limits of the Burgers-Huxley equations to the Burgers as well as Huxley equations are also discussed. Next, we consider the stationary Burgers-Huxley equation and establish the existence and uniqueness of weak solution by using a Faedo-Galerkin approximation technique and compactness arguments. Then, we discuss about the exponential stability of stationary solutions. Concerning numerical studies, we first derive error estimates for the semidiscrete Galerkin approximation. Finally, we present two computational examples to show the convergence numerically.

1D Generalised Burgers-Huxley: Proposed Solutions Revisited and Numerical Solution Using FTCS and NSFD Methods

“…This is the case of the closed form solution provided in Wang et al [8] and Deng [19]. The question of well-posedness, existence, and uniqueness of the solution to the Burgers-Huxley Equation ( 7) has been recently reported by Mohan and Khan [24]. One classic test for possible closed form solution to any differential equation is the Painleve test, which informs us about the possible integrability of the differential equation.…”

Conforming, nonconforming and DG methods for the stationary generalized Burgers- Huxley equation

“…For δ = 1 (that is, for the classical Burgers-Huxley equation), we obtain a simpler condition than (2.11) for the uniqueness of weak solution. In this case, the estimate (2.17) becomes (see [21])…”

“…Here ν > 0 is the constant diffusion coefficient, α > 0 is the advection coefficient, and β > 0, δ ≥ 1, γ ∈ (0, 1) are model parameters modulating the interplay between non-standard nonlinear advection, diffusion, and nonlinear reaction (or applied current) contributions. The global solvability of the one-dimensional Burgers-Huxley equation has been recently established in [21]. In this paper we extend the analysis to the multi-dimensional case.…”

“…The Burgers-Huxley equation is a special type of nonlinear advection-diffusionreaction problems that are of importance in applications in mechanical engineering, material sciences, and neurophysiology. Some examples include, for instance, particle transport [24], dynamics of ferroelectric materials [32], action potential propagation in nerve fibers [29], wall motion in liquid crystals [30], and many others (see also [12,21] and the references therein).…”

Large deviation principle for occupation measures of stochastic generalized Burgers-Huxley equation

“…The stochastic generalized Burgers-Huxley equation (SGBH) equation describes a prototype model for describing the interaction between reaction mechanisms, convection effects and diffusion transports (cf. [26]). We consider the generalized Burgers-Huxley equation perturbed by a random forcing, which is a space-time white noise (or Brownian sheet), as ∂u(t, ξ) ∂t = ν ∂ 2 u(t, ξ) ∂ξ 2 − αu δ (t, ξ) ∂u(t, ξ) ∂ξ + βu(t, ξ)(1 − u δ (t, ξ))(u δ (t, ξ) − γ)…”