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

Manil T. Mohan; Arbaz Khan

doi:10.3934/dcdsb.2020270

Article Contents

2021, Volume 26, Issue 7: 3943-3988. Doi: 10.3934/dcdsb.2020270

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

Manil T. Mohan^, and
Arbaz Khan

Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand 247667, India

^*Corresponding author: Manil T. Mohan
^*Corresponding author: Manil T. Mohan

Received: December 31, 2019

Revised: June 30, 2020

Early access: September 2020

Published: July 2021

The first author is supported by DST-INSPIRE Faculty Award (IFA17-MA110)

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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.

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Mathematics Subject Classification: Primary: 35K58, 35Q35; Secondary: 65N30, 37L30, 35D30, 35D35.

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Figure 1. Plots of solution (Case I) for the fixed time step size $ k = 1/10 $ and the different values of spatial mesh size (a) $ h = 1/2 $; (b) $ h = 1/4 $; (c) $ h = 1/8 $; (d) $ h = 1/16 $ at $ t = 1 $

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Figure 2. Plots of solution (Case I) for the fixed spatial mesh size $ h = 1/32 $ and the different values of time step size (a) $ k = 1/2 $; (b) $ k = 1/6 $; (c) $ k = 1/10 $ at $ t = 1 $

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Figure 3. Plots of solution (Case II) for the fixed time step size $ k = 1/100 $ and the different values of the spatial mesh size (a)$ h = 1/2 $; (b) $ h = 1/4 $; (c) $ h = 1/8 $; (d) $ h = 1/16 $ at $ t = 1 $

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Figure 4. Plots of solution (Case II) for the fixed spatial mesh size $ h = 1/32 $ and the different values of time step size (a) $ k = 1/2 $; (b) $ k = 1/6 $; (c) $ k = 1/10 $ at $ t = 1 $

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