Nonlinear Parabolic Equations Research Articles

Numerical modeling of surface subsidence due to compaction of soil with fine inclusions

A mathematical model of filtration consolidation of an inhomogeneous soil mass was formed taking into account the change in the size of the area during the compaction process. The inhomogeneity is considered as the presence of fine inclusions (geobarriers) the physical and mechanical characteristics of which differ from those of the main soil. From a mathematical viewpoint, the model is described by a one-phase Stefan problem that has a kinematic boundary condition on the upper moving boundary as its component. The purpose of the research is to find out the effect of fine inclusion on the dynamics of subsidence of the soil surface in the process of compaction. The change in the dimensions of the solution area is physically determined by the change in the volume of the pores of the porous medium in the process of dissipating excess pressure. If the permeability of the geobarrier is low, it affects the dynamics of consolidation processes and, accordingly, the magnitude of subsidence. Finite element solutions of the initial-boundary value problem for the nonlinear parabolic equation in the heterogeneous region with the conjugation condition of non-ideal contact were found. Numerical time discretization methods, a method for determining the change in the position of the upper boundary at discrete moments of time, and an algorithm for determining the physical and mechanical characteristics of a porous medium depending on the degree of consolidation are given. A number of test examples were considered, and the effect of a thin inclusion on the dynamics of the change in the position of the upper boundary of the problem solution area was investigated.

THE BEHAVIOR OF THE SOLUTIONS OF NONLINEAR PARABOLIC EQUATIONS

Abstract. The purpose of this paper is to study the behavior of solution of the problem for degenerate nonlinear parabolic equations on the boundary of a domain. In this case, the method of obtaining a priori energy estimates of solutions to parabolic boundary problems is used to study generalized solutions growing in the vicinity of a boundary point (finite or infinitely distant). The method used differs from the technique of obtaining the corresponding results in a linear situation.

Interior Holder Regularity and Infinite Time Extinction to a doubly Nonlinear Degenerate Parabolic Equations of Signed Solutions

We study doubly nonlinear parabolic equation with sign changing solutions. We established the Holder regularity and the infinite time extinction of it within a parabolic domain. GANIT J. Bangladesh Math. Soc. 43.1 (2023) 75- 91

Interior regularity to the signed solution of the singular doubly nonlinear parabolic equations

We study doubly nonlinear parabolic equation with sign changing solutions. We establish the Hölder regularity of the singular parabolic equations within a parabolic domain. J. Bangladesh Acad. Sci. 47(2); 241-248: December 2023

On the existence of time-periodic solutions of nonlinear parabolic differential equations with nonlocal boundary conditions of the Bitsadze-Samarskii type

We study a nonlinear parabolic differential equation in a bounded multidimensional domain with nonlocal boundary conditions of the Bitsadze-Samarskii type. We prove existence theorems for a periodic in time generalized solution. Su cient conditions for the existence of generalized solutions contain either an algebraic ellipticity condition or an algebraic strong ellipticity condition for the auxiliary differential-difference operator.

On the structure of weak solutions of the Riemann problem for a degenerate nonlinear diffusion equation

An explicit form of weak solutions to the Riemann problem for a degenerate nonlinear parabolic equation with a piecewise constant diffusion coe cient is found. It is shown that the lines of phase transitions (free boundaries) correspond to the minimum point of some strictly convex and coercive function of a nite number of variables. A similar result is true for Stefan’s problem. In the limit, when the number of phases tends to in nity, there arises a variational formulation of self-similar solutions to the equation with an arbitrary nonnegative diffusion function.

C1,α-regularity for solutions of degenerate/singular fully nonlinear parabolic equations

We establish the interior C1,α-estimate for viscosity solutions of degenerate/singular fully nonlinear parabolic equationsut=|Du|γF(D2u)+fin Q1, where γ>−1 and f∈C(Q1)∩L∞(Q1). For this purpose, we prove the well-posedness of the regularized Cauchy-Dirichlet problem{ut=(1+|Du|2)γ/2F(D2u)in Q1u=φon ∂pQ1, where γ>−2. Our approach utilizes the Bernstein method with approximations in view of the difference quotient.

Mixed Problem for a Nonlinear Parabolic Equation with Involution

Mixed Problem for a Nonlinear Parabolic Equation with Involution

Reconstruction coefficient analysis of honeybee collapse due to pesticide contamination

In this paper we consider the inverse problems of identifying space-dependent coefficients of the mortality rate of the bees and the rate of contamination of the forager bees by pesticides. The model is described by a weakly coupled system of two reaction-diffusion equations for the spatial distribution of uncontaminated and contaminated foraging bees. Final time t = T observations of the density of uncontaminant and contaminant forager bees are used. We propose two approaches for studying the problems. The first one uses the overspecified information to transform the problems into non-linear parabolic equations involving the solution values at the final time. This allows us to prove, using fixed-point arguments, existence of solution to the inverse problems. The second study employs the concept of the quasi-solution to establish existence of solution to the inverse problems as minimizers of least-square cost functionals.

Numerical study of the impact of directed migration of non-indigenous species on invasion scenarios

A mathematical model of competition under conditions of biological invasion, written in the form of a system of nonlinear parabolic equations, is considered. The competition of two closely related species — resident and invader — is studied. The dynamics of populations in a heterogeneous area is determined by local interaction and diffusion. For the invader population, interspecific taxis and directed migration caused by heterogeneity of living conditions are taken into account. In computational experiments, sets of migration parameters corresponding to various invasion scenarios are determined. An analysis of the influence of initial distributions on competitive exclusion and coexistence of species is given.

The Cauchy Problem for a Doubly Nonlinear Parabolic Equation with Variable Density and Nonlinear Time-Dependent Absorption

The Cauchy Problem for a Doubly Nonlinear Parabolic Equation with Variable Density and Nonlinear Time-Dependent Absorption

The Quenching Solutions of a Singular Parabolic Equation

This article is dedicated to the study of the self-similar solutions of a nonlinear parabolic equation. More precisely, we consider the following uni-dimensional equation: (E) : ut(x, t) = (u m)_xx(x, t) − |x|^q u^−p (x, t), x ∈ R, t > 0, where m > 1, q > 1 and p > 0. Initially, we employed a fixed point theorem and an associated energy function to establish the existence of solutions. Subsequently, we derived some important results on the asymptotic behavior of solutions near the origin.

Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds

Abstract In this article, we study the long-time dynamical behavior of the solution for a class of semilinear edge-degenerate parabolic equations on manifolds with edge singularities. By introducing a family of potential well and compactness method, we reveal the dependence between the initial data and the long-time dynamical behavior of the solution. Specifically, we give the threshold condition for the initial data, which makes the solution exist globally or blowup in finite-time with subcritical, critical, and supercritical initial energy, respectively. Moreover, we also discussed the long-time behavior of the global solution, the estimate of blowup time, and blowup rate. Our results show that the relationship between the initial data and the long-time behavior of the solution can be revealed in the weighted Sobolev spaces for nonlinear parabolic equations on manifolds with edge singularities.

A deep-genetic algorithm (deep-GA) approach for high-dimensional nonlinear parabolic partial differential equations

We propose a new method, called a deep-genetic algorithm (deep-GA), to accelerate the performance of the so-called deep-BSDE method, which is a deep learning algorithm to solve high dimensional partial differential equations through their corresponding backward stochastic differential equations (BSDEs). Recognizing the sensitivity of the solver to the initial guess selection, we embed a genetic algorithm (GA) into the solver to optimize the selection. We aim to achieve faster convergence for the nonlinear PDEs on a broader interval than deep-BSDE. Our proposed method is applied to two nonlinear parabolic PDEs, i.e., the Black-Scholes (BS) equation with default risk and the Hamilton-Jacobi-Bellman (HJB) equation. We compare the results of our method with those of the deep-BSDE and show that our method provides comparable accuracy with significantly improved computational efficiency.

The Fourier-based dimensional reduction method for solving a nonlinear inverse heat conduction problem with limited boundary data

The objective of this article is to introduce a novel technique for computing numerical solutions to the nonlinear inverse heat conduction problem. This involves solving nonlinear parabolic equations with Cauchy data provided on one side Γ of the boundary of the computational domain Ω. The key step of our proposed method is the truncation of the Fourier series of the solution to the governing equation. The truncation technique enables us to derive a system of 1D ordinary differential equations. Then, we employ the well-known Runge–Kutta method to solve this system, which aids in addressing the nonlinearity and the lack of data on ∂Ω∖Γ. This new approach is called the Fourier-based dimensional reduction method. By converting the high-dimensional problem into a 1D problem, we achieve exceptional computational speed. Numerical results are provided to support the effectiveness of our approach.

Entropy dissipative higher order accurate positivity preserving time-implicit discretizations for nonlinear degenerate parabolic equations

We develop entropy dissipative higher order accurate local discontinuous Galerkin (LDG) discretizations coupled with Diagonally Implicit Runge–Kutta (DIRK) methods for nonlinear degenerate parabolic equations with a gradient flow structure. Using the simple alternating numerical flux, we construct DIRK-LDG discretizations that combine the advantages of higher order accuracy, entropy dissipation and proper long-time behavior. We theoretically prove the entropy dissipation of the implicit Euler-LDG discretization without any time-step restrictions when no positivity constraint is imposed. Next, in order to ensure the positivity of the numerical solution, we use the Karush–Kuhn–Tucker (KKT) limiter, which achieves a positive solution by coupling the positivity preserving KKT conditions with higher order accurate DIRK-LDG discretizations using Lagrange multipliers. In addition, mass conservation of the positivity-limited solution is ensured by imposing a mass conservation equality constraint to the KKT equations. Under a time step restriction, the unique solvability and entropy dissipation for implicit first order accurate in time, but higher order accurate in space, positivity-preserving LDG discretizations with periodic boundary conditions are proved, which provide a first theoretical analysis of the KKT limiter. Finally, numerical results demonstrate the higher order accuracy and entropy dissipation of the positivity-preserving DIRK-LDG discretizations for problems requiring a positivity limiter. In addition, we can observe from the numerical results that the implicit time-discrete methods alleviate the time-step restrictions needed for the stability of the numerical discretizations, which improves computational efficiency.

An energy formula for fully nonlinear degenerate parabolic equations in one spatial dimension

An energy formula for fully nonlinear degenerate parabolic equations in one spatial dimension

Stability and boundedness of regular solutions for a Sisko flow in an infinite annular porous space

The present article is intended to study the boundedness of solutions for an unsteady non-Newtonian flow, whose strain–stress relationship is provided by the Sisko fluid model. Such kind of flow can appear in different scenarios, but we make it particular to the field of magnetohydrodynamics, as it is a remarkable area of research in its own right. The ideas exposed in this work can be extended, in a similar manner, to a wide range of applications. To make our fluid further general, we use the Darcy’s law to characterize the porous space in which the fluid is flowing. We develop the boundedness criteria provided that the velocity w and the function g=−∂w/∂r satisfy w,g,∂g/∂r,∂2g/∂r2∈L2(0,T;BMO), where BMO means bounded mean oscillation space. For this purpose, we will consider energy estimates in Sobolev spaces and develop the boundedness criteria for the resulting unsteady parabolic nonlinear equation.

Expansion of positivity for doubly nonlinear parabolic equations and its application

Expansion of positivity for doubly nonlinear parabolic equations and its application

A sixth-order finite difference HWENO scheme for nonlinear degenerate parabolic equation

A sixth-order finite difference HWENO scheme for nonlinear degenerate parabolic equation