Nonlinear Parabolic Equations Research Articles

The Homotopy Perturbation Method for Solving Nonlocal Initial-Boundary Value Problems for Parabolic and Hyperbolic Partial Differential Equations

To obtain approximate-exact solutions to nonlocal initial-boundary value problems (IBVPs) of linear and nonlinear parabolic and hyperbolic partial differential equations (PDEs) subject to initial and nonlocal boundary conditions of integral type, the homotopy perturbation method (HPM) is utilized in this study. The HPM is used to solve the specified nonlocal IBVPs, which are then transformed into local Dirichlet IBVPs. Some examples demonstrate how accurate and efficient the HPM.

Upper semicontinuity of pullback D$\mathcal {D}$‐attractors for nonlinear parabolic equation with nonstandard growth condition

AbstractThis paper is devoted to the well‐posed problem and the existence of pullback ‐attractors for a class of nonlinear parabolic equation with nonstandard growth condition. First, by making use of Galerkin's method and monotone operator method, the existence of solutions is proved in Orlicz–Sobolev space with variable exponents depending on time and space, then the uniqueness and continuity of solutions are also obtained. Finally, by verifying the pullback ‐asymptotic compactness, the existence of pullback ‐attractors is proved. In particular, the upper semicontinuity of pullback ‐attractors of the corresponding equation with respect to the disturbance parameter λ is also proved.

L 2 norm error estimates of BDF methods up to fifth-order for the phase field crystal model

Abstract The well-known backward difference formula (BDF) of the third, the fourth and the fifth orders are investigated for time integration of the phase field crystal model. By building up novel discrete gradient structures of the BDF-$\textrm{k}$ ($\textrm{k}=3,4,5$) formulas, we establish energy dissipation laws at the discrete levels and then obtain a priori solution estimates for the associated numerical schemes; however, we can not build any discrete energy dissipation law for the corresponding BDF-6 scheme because the BDF-6 formula itself does not have any discrete gradient structures. With the help of the discrete orthogonal convolution kernels and Young-type convolution inequalities, some concise $L^{2}$ norm error estimates (with respect to the starting data in the $L^{2}$ norm) are established via the discrete energy technique. To the best of our knowledge, this is the first time such type $L^{2}$ norm error estimates of non-A-stable BDF schemes are obtained for nonlinear parabolic equations. Numerical examples are presented to verify and support the theoretical analysis.

Equivalence of "generalized" solutions for nonlinear parabolic equations with variable exponents and diffuse measure data

We prove the equivalence of suitably defined weak solutions of a nonhomogeneous initial-boundary value problem for a class of nonlinear parabolic equations. We also develop the notion of both "renormalized" and "entropy" solutions with respect to the "generalized" $p(\cdot)$-capacity, initial datum, and diffuse measure data (which does not charge the set of null $p(\cdot)$-capacity). Conditions, under which "generalized weak" solutions of the nonhomogeneous problem are in fact well-defined, are also given.

On the theory of entropy suband supersolutions of nonlinear degenerate parabolic equations

We consider a second-order nonlinear degenerate anisotropic parabolic equation in the case when the flux vector is only continuous and the nonnegative diffusion matrix is bounded and measurable. The concepts of entropy sub- and supersolution of the Cauchy problem are introduced, so that the entropy solution of this problem, understood in the sense of Chen-Perthame, is both an entropy sub- and supersolution. It is established that the maximum of entropy subsolutions of the Cauchy problem is also an entropy subsolution of this problem. This result is used to prove the existence of the largest entropy subsolution (and the smallest entropy supersolution). It is also shown that the largest entropy subsolution and the smallest entropy supersolution are also entropy solutions.

Calderón-Zygmund estimates for nonlinear parabolic equations with matrix weights on nonsmooth domains

A wider class of nonlinear parabolic equations with degenerate/singular matrix weights is studied for Calderón-Zygmund type estimates of weak solutions in the setting of weighted Sobolev spaces. Minimal regularity assumptions on the associated nonlinearities as well as weights are investigated for such classical estimates.

Continuity of the Temperature in a Multi-Phase Transition Problem. Part III

We establish local continuity of locally bounded weak solutions to a doubly nonlinear parabolic equation that models the temperature in multi-phase transitions. The enthalpy allows for general maximal monotone graphs of the temperature. Remarkably, moduli of continuity can be estimated without an explicit form of the enthalpy.

Solvability of nonlinear pseudo‐parabolic equations involving generalized Caputo fractional derivatives

This study considers nonlinear fractional pseudo parabolic equations, which include the generalized Caputo fractional derivatives of a function with respect to an appropriate function, with general nonlocal initial conditions. Here, the fractional derivative is generalized from many well‐known ones, such as the Caputo, Caputo–Katugampola, Caputo–Hadamard, Erdélyi–Kober, and Liouville–Caputo derivatives. We propose sufficient conditions to ensure that the problem has at least one or a unique mild solution. Furthermore, we investigate the continuous dependence of the mild solutions on the fractional order and other inputs. Particularly, source functions in this study may have temporal singularities. Finally, we provide numerical experiments to illustrate and confirm our theoretical findings.

Infinite Horizon Optimal Control for a General Class of Semilinear Parabolic Equations

A class of infinite horizon optimal control problems subject to semilinear parabolic equations is investigated. First and second order optimality conditions are obtained, in the presence of constraints on the controls, which can be either pointwise in space-time, or pointwise in time and L^2 in space. These results rely on a new L^infty estimate for nonlinear parabolic equations in an essential manner.

Elliptic gradient estimates for a nonlinear equation with Dirichlet boundary condition

We prove local elliptic type gradient estimates for a nonlinear parabolic equation (including the fast diffusion equation and the porous medium equation) with Dirichlet boundary condition on Riemannian manifolds with compact boundary when the Ricci curvature and the mean curvature of boundary are bounded below. As an application, we obtain Liouville type results for certain ancient solutions to the nonlinear parabolic equation on Riemannian manifolds with compact boundary.

On the Solvability of Nonlinear Parabolic Functional-Differential Equations with Shifts in the Spatial Variables

On the Solvability of Nonlinear Parabolic Functional-Differential Equations with Shifts in the Spatial Variables

A selfdual variational principle with minimal hypothesis and applications to stationary, dynamic and stochastic equations

We revisit the self-dual variational approach to the resolution of non-linear stationary, parabolic and stochastic partial differential equations involving a monotone non-linearity, by relaxing the hypothesis under which it is applicable, in two directions: The prohibitive coercivity condition which was problematic for applications to evolution equations, be them deterministic or stochastic, and the restrictive “regularity” property that required the domain of the non-linear operator under study to be large. The new principle is applied to give more direct proofs for the existence of weak solutions for incompressible Navier-Stokes equations in their stationary, dynamic or stochastic versions.

Dynamic Fuzzy Boundary Output Feedback Control for Nonlinear Delayed Parabolic Partial Differential Equation Systems Under Noncollocated Boundary Measurement

For nonlinear space-varying parabolic partial differential equation systems (PPDESs) with random time-varying delay, this paper introduces a dynamic fuzzy boundary output feedback (DFBOF) control under non-collocated boundary measurement (NCBM). Initially, the nonlinear delayed PPDESs are represented by Takagi-Sugeno (T–S) fuzzy models and random time-varying delay is considered by taking the influence of uncertain factors, which belongs to two intervals in a probabilistic way. Since the system state is not fully available and NCBM makes the boundary control design very difficult, a fuzzy observer under NCBM is presented to surmount the design difficulty. Subsequently, an observer-based fuzzy boundary controller is proposed and spatial linear matrix inequality (SLMI) based sufficient conditions to ensure mean square exponential stability are obtained for closed-loop delayed PPDESs by utilizing the Lyapunov direct method and Wirtinger inequality. Then, to solve the SLMIs, the feasibility conditions of DFBOF controller design for nonlinear delayed PPDES are expressed in LMIs. Lastly, two examples are offered to demonstrate the validity of the presented dynamic fuzzy boundary control approach.

System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

Harnack Estimation for Nonlinear, Weighted, Heat-Type Equation along Geometric Flow and Applications

The method of gradient estimation for the heat-type equation using the Harnack quantity is a classical approach used for understanding the nature of the solution of these heat-type equations. Most of the studies in this field involve the Laplace–Beltrami operator, but in our case, we studied the weighted heat equation that involves weighted Laplacian. This produces a number of terms involving the weight function. Thus, in this article, we derive the Harnack estimate for a positive solution of a weighted nonlinear parabolic heat equation on a weighted Riemannian manifold evolving under a geometric flow. Applying this estimation, we derive the Li–Yau-type gradient estimation and Harnack-type inequality for the positive solution. A monotonicity formula for the entropy functional regarding the estimation is derived. We specify our results for various different flows. Our results generalize some works.

Strongly nonlinear parabolic equations with natural growth in general domains

Strongly nonlinear parabolic equations with natural growth in general domains

Diffusion - Convection Equation of Galactic Cosmic Rays (GCR) in the Atmosphere and Its Analytical, Numerical Solutions by Using Finite Elements Method Using Parker Transport Equation

This study depicts that to find the Analytical and numerical solutions of Diffusion – Convections Equations of Galactic Cosmic Rays (GCRs) by Finite Element method (FEM) and also to find the Energy Equation of GCRs by using a part of Parker’s transport equation. This considers moreover centres on the exactness and acknowledgment of the FEM strategy by utilizing dissemination blunder, scattering mistake and add up to blunder investigation. The comes about are depicted both graphically and in a unthinkable frame, which essentially guarantees the method’s legitimacy and the algorithm’s proficiency to maintain the exactness, effortlessness, and nonlinear Convection-Diffusion Equation conditions. The proposed method may be connected for tackling any nonlinear convection diffusion equation. We concentrate on assaying the confluence and stability of the nonlinear parabolic partial differential equation. This study focuses on the delicacy and acceptance of FEM method by exercising dispersion error dissipation error, and total error analysis.

A sixth-order central WENO scheme for nonlinear degenerate parabolic equations

In this paper, we develop a new sixth-order finite difference central weighted essentially non-oscillatory (WENO) scheme with Z-type nonlinear weights for nonlinear degenerate parabolic equations. The centered polynomial is introduced for the WENO reconstruction to avoid the negative linear weights. We choose the Z-type nonlinear weights based on the $$L^2$$ -norm smoothness indicators, yielding the new WENO scheme with less computational cost. It is also confirmed that the proposed central WENO scheme with the devised nonlinear weights achieves sixth order accuracy in smooth regions. One- and two-dimensional numerical examples are presented to demonstrate the improved performance of the proposed central WENO scheme.

Nonlinear Parabolic Equations with Singular Coefficient and with Variable Exponents

Nonlinear Parabolic Equations with Singular Coefficient and with Variable Exponents

Spline collocation methods for solving some types of nonlinear parabolic partial differential equations

Spline collocation methods for solving some types of nonlinear parabolic partial differential equations