Burgers–Huxley Equation with Memory: Models & Analysis

• The generalized Burgers–Huxley equation with memory is a nonlinear model that incorporates hereditary effects via a convolution kernel acting on the diffusion term.
• Analytical frameworks using finite element, discontinuous Galerkin, and hp-DG methods establish existence, uniqueness, and optimal error estimates under various boundary and kernel conditions.
• Stabilization and feedback control techniques, including Riccati-based methods, are integrated to manage pattern formation and ensure energy decay in numerical simulations.

Searching arXiv for papers on the generalized Burgers–Huxley equation with memory and related numerical/stabilization analyses. The generalized Burgers–Huxley equation with memory is a nonlinear advection–diffusion–reaction model in which the diffusive part is augmented by a hereditary convolution term, typically acting on the Laplacian through either a weakly singular kernel or an exponentially decaying fading-memory kernel. In the recent arXiv literature, the model appears in finite element, discontinuous Galerkin, hp-time-stepping, a posteriori estimation, and feedback-stabilization settings, with homogeneous Dirichlet, mixed Dirichlet–Neumann, and controlled boundary conditions treated in different formulations (Mahajan et al., 2023, Mahajan et al., 2024, Akram et al., 27 Mar 2025, Bag et al., 4 Aug 2025).

1. Governing equations and memory mechanisms

A standard weakly singular-kernel formulation on a bounded domain $\Omega\subset\mathbb{R}^d$, $d\in\{2,3\}$, is

$$\begin{aligned} \partial_t u(x,t) + \alpha\,u(x,t)^{\delta}\sum_{i=1}^d \partial_{x_i}u(x,t) - \nu\,\Delta u(x,t) -\eta \int_0^t K(t-\tau)\,\Delta u(x,\tau)\,d\tau \ = \beta\,u(x,t)\,\big(1-u(x,t)^{\delta}\big)\,\big(u(x,t)^{\delta}-\gamma\big) + f(x,t), \qquad (x,t)\in\Omega\times(0,T), \end{aligned}$$

with homogeneous Dirichlet boundary data and initial condition $u(x,0)=u_0(x)$. Here $\alpha,\nu,\beta\ge 0$, $\eta\ge 0$, $\delta\ge 1$, and $\gamma\in(0,1)$. The memory term is a Volterra convolution in time applied to $\Delta u$, so the hereditary effect is through diffusion rather than through a discrete delay state $u(x,t-\tau)$ (Mahajan et al., 2024, Mahajan et al., 2023).

The associated stationary generalized Burgers–Huxley equation removes both $d\in\{2,3\}$0 and the memory contribution: $d\in\{2,3\}$1

For weak formulations, the nonlinear structures are often written through

$d\in\{2,3\}$2

In this notation, the memory term becomes a time convolution on gradients after integration by parts: $d\in\{2,3\}$3

Two kernel classes dominate the literature. For weakly singular memory, one prototypical choice is

$d\in\{2,3\}$4

or, in computational studies, $d\in\{2,3\}$5. For stabilization problems, the kernel is typically exponential: $d\in\{2,3\}$6 which is positive, strictly decaying, integrable, completely monotone, and admits the explicit Laplace transform $d\in\{2,3\}$7 in the boundary-control analysis (Mahajan et al., 2024, Akram et al., 27 Mar 2025, Bag et al., 4 Aug 2025).

2. Variational structure, positivity, and regularity

The basic analytical framework uses the Gelfand triplet $d\in\{2,3\}$8, augmented by $d\in\{2,3\}$9 control to handle the Huxley nonlinearity. For the time-dependent weakly singular-kernel problem, one seeks

$$\begin{aligned} \partial_t u(x,t) + \alpha\,u(x,t)^{\delta}\sum_{i=1}^d \partial_{x_i}u(x,t) - \nu\,\Delta u(x,t) -\eta \int_0^t K(t-\tau)\,\Delta u(x,\tau)\,d\tau \ = \beta\,u(x,t)\,\big(1-u(x,t)^{\delta}\big)\,\big(u(x,t)^{\delta}-\gamma\big) + f(x,t), \qquad (x,t)\in\Omega\times(0,T), \end{aligned}$$0

with $$\begin{aligned} \partial_t u(x,t) + \alpha\,u(x,t)^{\delta}\sum_{i=1}^d \partial_{x_i}u(x,t) - \nu\,\Delta u(x,t) -\eta \int_0^t K(t-\tau)\,\Delta u(x,\tau)\,d\tau \ = \beta\,u(x,t)\,\big(1-u(x,t)^{\delta}\big)\,\big(u(x,t)^{\delta}-\gamma\big) + f(x,t), \qquad (x,t)\in\Omega\times(0,T), \end{aligned}$$1 in a dual space, and tests against $$\begin{aligned} \partial_t u(x,t) + \alpha\,u(x,t)^{\delta}\sum_{i=1}^d \partial_{x_i}u(x,t) - \nu\,\Delta u(x,t) -\eta \int_0^t K(t-\tau)\,\Delta u(x,\tau)\,d\tau \ = \beta\,u(x,t)\,\big(1-u(x,t)^{\delta}\big)\,\big(u(x,t)^{\delta}-\gamma\big) + f(x,t), \qquad (x,t)\in\Omega\times(0,T), \end{aligned}$$2 (Mahajan et al., 2024).

A structural assumption common to the weakly singular analyses is that $$\begin{aligned} \partial_t u(x,t) + \alpha\,u(x,t)^{\delta}\sum_{i=1}^d \partial_{x_i}u(x,t) - \nu\,\Delta u(x,t) -\eta \int_0^t K(t-\tau)\,\Delta u(x,\tau)\,d\tau \ = \beta\,u(x,t)\,\big(1-u(x,t)^{\delta}\big)\,\big(u(x,t)^{\delta}-\gamma\big) + f(x,t), \qquad (x,t)\in\Omega\times(0,T), \end{aligned}$$3 is of positive type: $$\begin{aligned} \partial_t u(x,t) + \alpha\,u(x,t)^{\delta}\sum_{i=1}^d \partial_{x_i}u(x,t) - \nu\,\Delta u(x,t) -\eta \int_0^t K(t-\tau)\,\Delta u(x,\tau)\,d\tau \ = \beta\,u(x,t)\,\big(1-u(x,t)^{\delta}\big)\,\big(u(x,t)^{\delta}-\gamma\big) + f(x,t), \qquad (x,t)\in\Omega\times(0,T), \end{aligned}$$4 This positivity is repeatedly used in energy identities and stability estimates. In the a posteriori DG analysis, positivity of type and integrability near $$\begin{aligned} \partial_t u(x,t) + \alpha\,u(x,t)^{\delta}\sum_{i=1}^d \partial_{x_i}u(x,t) - \nu\,\Delta u(x,t) -\eta \int_0^t K(t-\tau)\,\Delta u(x,\tau)\,d\tau \ = \beta\,u(x,t)\,\big(1-u(x,t)^{\delta}\big)\,\big(u(x,t)^{\delta}-\gamma\big) + f(x,t), \qquad (x,t)\in\Omega\times(0,T), \end{aligned}$$5 are sufficient to control the memory contribution, and no fractional Grönwall inequality is required (Mahajan et al., 2024). In the conforming and nonconforming finite element analyses, the same positivity enters the continuous and discrete energy inequalities (Mahajan et al., 2023, Mahajan et al., 2023).

Existence, uniqueness, and regularity are established under standard data assumptions. For weak solutions, one result assumes $$\begin{aligned} \partial_t u(x,t) + \alpha\,u(x,t)^{\delta}\sum_{i=1}^d \partial_{x_i}u(x,t) - \nu\,\Delta u(x,t) -\eta \int_0^t K(t-\tau)\,\Delta u(x,\tau)\,d\tau \ = \beta\,u(x,t)\,\big(1-u(x,t)^{\delta}\big)\,\big(u(x,t)^{\delta}-\gamma\big) + f(x,t), \qquad (x,t)\in\Omega\times(0,T), \end{aligned}$$6, $$\begin{aligned} \partial_t u(x,t) + \alpha\,u(x,t)^{\delta}\sum_{i=1}^d \partial_{x_i}u(x,t) - \nu\,\Delta u(x,t) -\eta \int_0^t K(t-\tau)\,\Delta u(x,\tau)\,d\tau \ = \beta\,u(x,t)\,\big(1-u(x,t)^{\delta}\big)\,\big(u(x,t)^{\delta}-\gamma\big) + f(x,t), \qquad (x,t)\in\Omega\times(0,T), \end{aligned}$$7, and $$\begin{aligned} \partial_t u(x,t) + \alpha\,u(x,t)^{\delta}\sum_{i=1}^d \partial_{x_i}u(x,t) - \nu\,\Delta u(x,t) -\eta \int_0^t K(t-\tau)\,\Delta u(x,\tau)\,d\tau \ = \beta\,u(x,t)\,\big(1-u(x,t)^{\delta}\big)\,\big(u(x,t)^{\delta}-\gamma\big) + f(x,t), \qquad (x,t)\in\Omega\times(0,T), \end{aligned}$$8 of positive type; the resulting solution belongs to

$$\begin{aligned} \partial_t u(x,t) + \alpha\,u(x,t)^{\delta}\sum_{i=1}^d \partial_{x_i}u(x,t) - \nu\,\Delta u(x,t) -\eta \int_0^t K(t-\tau)\,\Delta u(x,\tau)\,d\tau \ = \beta\,u(x,t)\,\big(1-u(x,t)^{\delta}\big)\,\big(u(x,t)^{\delta}-\gamma\big) + f(x,t), \qquad (x,t)\in\Omega\times(0,T), \end{aligned}$$9

with

$u(x,0)=u_0(x)$0

and $u(x,0)=u_0(x)$1 (Mahajan et al., 2023). Uniqueness is shown either under $u(x,0)=u_0(x)$2 or under the parameter condition

$u(x,0)=u_0(x)$3

For stronger regularity, if $u(x,0)=u_0(x)$4 is convex or has $u(x,0)=u_0(x)$5-boundary, $u(x,0)=u_0(x)$6, and $u(x,0)=u_0(x)$7, then

$u(x,0)=u_0(x)$8

Under $u(x,0)=u_0(x)$9 and $\alpha,\nu,\beta\ge 0$0, one further obtains

$\alpha,\nu,\beta\ge 0$1

and with $\alpha,\nu,\beta\ge 0$2,

$\alpha,\nu,\beta\ge 0$3

The stabilization literature with exponential kernels uses related weak and strong solution spaces, but under mixed boundary conditions and, in the boundary-control case, nonhomogeneous Dirichlet data on $\alpha,\nu,\beta\ge 0$4 and homogeneous Neumann data on $\alpha,\nu,\beta\ge 0$5 (Mahajan et al., 2023, Bag et al., 4 Aug 2025).

3. Discretization frameworks

The numerical literature on the GBHE with memory covers conforming FEM, nonconforming Crouzeix–Raviart FEM, SIPG-type DGFEM in space, backward Euler and Crank–Nicolson time discretizations, and hp-DG time-stepping (Mahajan et al., 2023, Mahajan et al., 2023, Mahajan et al., 2024, Mahajan et al., 2024).

Formulation	Space discretization	Time treatment
Conforming approximation	$\alpha,\nu,\beta\ge 0$6, typically $\alpha,\nu,\beta\ge 0$7	semi-discrete, backward Euler, or hp-DG
Nonconforming approximation	Crouzeix–Raviart space $\alpha,\nu,\beta\ge 0$8	backward Euler with positive memory quadrature
DG approximation	SIPG / DGFEM space $\alpha,\nu,\beta\ge 0$9 or $\eta\ge 0$0	semi-discrete, backward Euler, Crank–Nicolson, or hp-DG

For conforming FEM, the semi-discrete scheme seeks $\eta\ge 0$1 such that

$\eta\ge 0$2

for all $\eta\ge 0$3 (Mahajan et al., 2023).

For nonconforming CR methods, the central device is a skew-symmetrized convection form

$\eta\ge 0$4

which satisfies $\eta\ge 0$5. This identity is used to obtain stability without parameter restrictions (Mahajan et al., 2023).

For DG methods in space, the diffusion term is typically discretized by the symmetric interior penalty form

$\eta\ge 0$6

with DG norm

$\eta\ge 0$7

or, in the adaptive a posteriori analysis,

$\eta\ge 0$8

The DG advection form is built with upwind fluxes and satisfies $\eta\ge 0$9 and $\delta\ge 1$0 in the analyses that exploit energy cancellation (Mahajan et al., 2024, Mahajan et al., 2023).

Time discretization of memory is usually handled by product integration. In backward Euler form,

$\delta\ge 1$1

with

$\delta\ge 1$2

The fully discrete DG a posteriori paper additionally treats Crank–Nicolson with midpoint quadrature and mesh-transfer operators $\delta\ge 1$3, and states that no CFL restriction is imposed in the analysis; stability follows from positivity of the kernel and elliptic coercivity of the DG form (Mahajan et al., 2024).

The hp-DG time-stepping formulation introduces a time partition $\delta\ge 1$4 and piecewise polynomial spaces

$\delta\ge 1$5

with jumps $\delta\ge 1$6. In the case $\delta\ge 1$7, the method reduces to a backward-Euler-type update (Mahajan et al., 2024).

4. Error analysis: a priori, a posteriori, and optimality

The weakly singular-kernel literature contains both a priori and a posteriori error analyses. In conforming FEM, a semi-discrete error estimate under minimal regularity reads

$\delta\ge 1$8

and the fully discrete backward-Euler scheme satisfies

$\delta\ge 1$9

If $\gamma\in(0,1)$0 with $\gamma\in(0,1)$1, then $\gamma\in(0,1)$2, and the temporal error becomes $\gamma\in(0,1)$3 (Mahajan et al., 2023).

For nonconforming CR and DG spatial discretizations, semi-discrete error estimates of the form

$\gamma\in(0,1)$4

are derived, together with fully discrete bounds containing

$\gamma\in(0,1)$5

for CR and the analogous DG estimate with the DG energy norm (Mahajan et al., 2023).

The hp-DG time-stepping analysis establishes energy-norm optimality with explicit dependence on local time-step sizes $\gamma\in(0,1)$6, polynomial degrees $\gamma\in(0,1)$7, spatial mesh sizes $\gamma\in(0,1)$8, and spatial degrees $\gamma\in(0,1)$9. For analytic solutions, the local projection estimates yield exponential convergence in $\Delta u$0; for algebraic regularity, the estimates involve the factor

$\Delta u$1

In the fully discrete conforming and DG settings, the resulting bounds are optimal in the energy norm with respect to $\Delta u$2, while the $\Delta u$3-in-time error is suboptimal by one power of $\Delta u$4; for DG in space, the $\Delta u$5-dependence is also suboptimal relative to the conforming case (Mahajan et al., 2024).

The a posteriori analysis for DG formulations introduces residual-based estimators for the stationary problem, the semi-discrete problem with memory, and fully discrete backward Euler and Crank–Nicolson schemes. For the stationary GBHE, local residuals are

$\Delta u$6

with flux-jump residuals $\Delta u$7 and jump terms $\Delta u$8. The global estimator $\Delta u$9 satisfies

$u(x,t-\tau)$0

For the semi-discrete memory problem,

$u(x,t-\tau)$1

where $u(x,t-\tau)$2 contains cell residuals, face residuals, jump terms, and explicit memory-jump contributions. For fully discrete backward Euler and Crank–Nicolson schemes, the reliability bounds combine time indicators $u(x,t-\tau)$3, spatial indicators $u(x,t-\tau)$4, and kernel-quadrature oscillation terms $u(x,t-\tau)$5 (Mahajan et al., 2024).

The same paper proves optimal $u(x,t-\tau)$6-error estimates for the stationary and semi-discrete evolutionary problems. In particular, for DG degree $u(x,t-\tau)$7,

$u(x,t-\tau)$8

for the stationary problem, and for the semi-discrete time-dependent problem

$u(x,t-\tau)$9

Fully discrete $d\in\{2,3\}$00-rates in $d\in\{2,3\}$01 are not proved there, but numerical results report first-order temporal convergence for backward Euler and second-order temporal convergence for Crank–Nicolson (Mahajan et al., 2024).

5. Feedback stabilization and control-theoretic formulations

A separate line of work studies GBHEs with exponential memory kernels from the viewpoint of stabilizability around zero or non-constant steady states. One formulation on a bounded $d\in\{2,3\}$02 domain with homogeneous Dirichlet boundary conditions and distributed interior control is

$d\in\{2,3\}$03

Introducing

$d\in\{2,3\}$04

converts the memory equation into a coupled system amenable to semigroup and Riccati analysis (Akram et al., 27 Mar 2025).

For the linearized principal system around a steady state, the state $d\in\{2,3\}$05 satisfies

$d\in\{2,3\}$06

where $d\in\{2,3\}$07 generates an analytic semigroup on $d\in\{2,3\}$08. The spectral analysis produces explicit eigenvalues

$d\in\{2,3\}$09

with $d\in\{2,3\}$10, and all eigenvalues satisfy $d\in\{2,3\}$11. After a spectral shift $d\in\{2,3\}$12, a feedback operator is obtained from the algebraic Riccati equation

$d\in\{2,3\}$13

leading to the closed-loop generator $d\in\{2,3\}$14, which is exponentially stable. The nonlinear stabilization results are then obtained by Banach fixed point arguments in the solution space

$d\in\{2,3\}$15

with smallness assumptions on $d\in\{2,3\}$16 and, for non-constant steady states, on $d\in\{2,3\}$17 (Akram et al., 27 Mar 2025).

The boundary-control literature treats mixed boundary conditions

$d\in\{2,3\}$18

and a fading memory term

$d\in\{2,3\}$19

After shifting by a decay factor $d\in\{2,3\}$20 and performing an elliptic lifting $d\in\{2,3\}$21, the feedback is constructed from the eigenfunctions $d\in\{2,3\}$22 of the Laplacian with Dirichlet conditions on $d\in\{2,3\}$23 and Neumann conditions on $d\in\{2,3\}$24. Under the linear independence hypothesis

$d\in\{2,3\}$25

the finite-dimensional Dirichlet boundary controller is

$d\in\{2,3\}$26

The linear closed-loop system satisfies

$d\in\{2,3\}$27

and the full nonlinear system is stabilized by the same controller through a Banach fixed point theorem on

$d\in\{2,3\}$28

If the steady state is zero, the additional constraint $d\in\{2,3\}$29 is not needed (Bag et al., 4 Aug 2025).

These stabilization results concern exponential kernels rather than weakly singular kernels. A plausible implication is that the explicit auxiliary-variable reformulation $d\in\{2,3\}$30 and the resolvent-based spectral analysis are especially well aligned with fading-memory kernels of exponential type.

6. Numerical behavior, adaptive refinement, and current limitations

The numerical studies cover manufactured solutions, singular solutions, adaptive refinement, finite-dimensional feedback simulation, and qualitative pattern-formation problems (Mahajan et al., 2024, Akram et al., 27 Mar 2025, Mahajan et al., 2023, Mahajan et al., 2023, Mahajan et al., 2024).

For weakly singular kernels, conforming, nonconforming, and DG discretizations on $d\in\{2,3\}$31 with $d\in\{2,3\}$32 repeatedly show first-order spatial convergence for piecewise linear spatial approximation in the relevant energy norm. In the conforming approximation paper, smooth-kernel tests with $d\in\{2,3\}$33 also exhibit first-order spatial convergence in 2D and 3D, both with memory $d\in\{2,3\}$34 and without memory $d\in\{2,3\}$35 (Mahajan et al., 2023). The hp-DG time-stepping paper reports, for $d\in\{2,3\}$36 in time, the expected increase in order, including $d\in\{2,3\}$37 errors of order approximately $d\in\{2,3\}$38 and $d\in\{2,3\}$39 and $d\in\{2,3\}$40 errors of order approximately $d\in\{2,3\}$41 and $d\in\{2,3\}$42 in the quadratic-space tests, as well as analogous optimal rates in 3D and in Caputo-type fractional experiments (Mahajan et al., 2024).

The adaptive a posteriori paper applies the standard loop

$d\in\{2,3\}$43

with a max-strategy marking criterion

$d\in\{2,3\}$44

On L-shaped domains, adaptive refinement concentrates near steep localized gradients or re-entrant corners. For the singular solution $d\in\{2,3\}$45, uniform meshes are suboptimal, but adaptive refinement recovers optimal rates. A time-varying singularity test without memory $d\in\{2,3\}$46 shows that a backward-Euler-plus-DG adaptive algorithm tracks a moving peak by refining around the moving singularity at each time step (Mahajan et al., 2024).

The stabilization papers include finite element simulations on $d\in\{2,3\}$47 with $d\in\{2,3\}$48 FEM semi-discretization. In the Riccati-based interior-control setting, the discrete algebraic Riccati equation is solved and the closed-loop eigenvalues are moved into the negative half-plane; the zero steady state and the non-constant steady state $d\in\{2,3\}$49 both exhibit energy decay under feedback, whereas the uncontrolled shifted dynamics do not (Akram et al., 27 Mar 2025).

Several application-oriented computations treat coupled or phenomenological systems. One conforming FEM study considers a FitzHugh–Nagumo-type system with memory and reports that spiral-wave dynamics respond to the nonlinearity exponent $d\in\{2,3\}$50 while the weakly singular memory modulates pattern formation (Mahajan et al., 2023). The nonconforming/DG paper also examines a fractional-time-derivative extension and 2D spiral-wave behavior on $d\in\{2,3\}$51, reporting that small $d\in\{2,3\}$52 preserves spiral structure while larger $d\in\{2,3\}$53 affects or reverses it (Mahajan et al., 2023). The hp-DG paper includes a prey–predator application in which increasing memory prolongs dynamics and affects attractors (Mahajan et al., 2024).

Several limitations are explicit. The weakly singular-kernel analyses generally assume $d\in\{2,3\}$54 and positive type; stronger singularities or nonpositive kernels are not covered in the conforming, nonconforming, and a posteriori works (Mahajan et al., 2023, Mahajan et al., 2023, Mahajan et al., 2024). Most nonlinear estimates are restricted to $d\in\{2,3\}$55, with upper bounds on the exponent in three dimensions such as $d\in\{2,3\}$56 or $d\in\{2,3\}$57 in the stabilization setting (Akram et al., 27 Mar 2025, Bag et al., 4 Aug 2025). The fully discrete analysis in the adaptive DG paper is a posteriori rather than a priori in $d\in\{2,3\}$58, even though the observed temporal orders match first-order backward Euler and second-order Crank–Nicolson (Mahajan et al., 2024). In the same work, the discrete memory sum requires a growing history at each time step, so the cost and storage are $d\in\{2,3\}$59 without compression; fast convolution or sum-of-exponentials acceleration is noted as beyond the scope (Mahajan et al., 2024).

A recurrent misconception is that the term “delayed GBHE” denotes a discrete delay equation. In the finite element papers, it does not: no term of the form $d\in\{2,3\}$60 appears, and the entire delay mechanism is hereditary memory represented by a time convolution with $d\in\{2,3\}$61 acting on $d\in\{2,3\}$62 or $d\in\{2,3\}$63 (Mahajan et al., 2023, Mahajan et al., 2023).