Well-Posedness and Numerical Study for Solutions of a Parabolic Equation with Variable-Exponent Nonlinearities

In this paper, modified variational iteration algorithm-II is investigated for finding approximate solutions of nonlinear Parabolic equations. Comparisons of the MVIA-II with trigonometric B-spline collocation method, variational iteration method, homotopy perturbation transform method, Adomian decomposition method, and modified variational iteration method are carried out, which show that the proposed algorithm (MVIA-II) is robust one. Some nonlinear Parabolic equations are given to demonstrate the implementation and accuracy of the MVIA-II.

Theorems on the existence and uniqueness of the statistical solutions and for the existence of stationary solutions for nonlinear stochastic equations are proved. Markov families of statistical solutions are constructed. The ergodic properties of monotone systems are investigated. A Girsanov type theorem on the absolute continuity of the statistical solution of a nonlinear stochastic parabolic equation with respect to the statistical solution of the monotone equation is proved.

We study the Cauchy problem and the initial boundary value problem (IBVP) for nonlinear parabolic equations in $\\mathcal{C}_b([0,T);L^p)$ and $L^q(0,T;L^p)$.\nWe give a unified method to construct local mild solutions of the Cauchy problem or IBVP for a class of nonlinear parabolic equations in $\\mathcal{C}_b([0,T);L^p)$ or $L^q(0,T;L^p)$ by introducing admissible triplet, generalized admissible triplet and establishing time space estimates for the solutions to the linear parabolic equations.\nMoreover, using our method, we also obtain the existence of global small solutions to the nonlinear parabolic equations.

Vanishing solutions of anisotropic parabolic equations with variable nonlinearity

In this paper, our primary focus is on studying the properties of self-similar solutions to the Cauchy problem. We specifically examine the behavior of these solutions in a double nonlinear time-dependent parabolic equation and their absorption in a non-homogeneous medium. Through the research of the topic matter, our aim is to deliver a more thorough comprehension of the finite speed perturbations propagation in the solution of the Cauchy problem for a nonlinear parabolic equation. This establishment of a property is essential to understand the dynamic nature of these equations. Furthermore, we delve into the self-similar analysis of the solution, which allows us to ascertain the condition of Fujita type global solvability for the Cauchy problem in a double nonlinear degenerate-type parabolic equation within a non-homogeneous medium. This analysis provides valuable insights into the behavior and potential solvability of these equations on a global mean. Additionally, we establish estimates for weak solutions depending on the growing density and the value of numerical parameters. By establishing these estimates, we provide a more comprehensive understanding of the behavior of the solutions in different scenarios.

Porous medium equation with absorption and a nonlinear boundary condition

Necessary conditions of optimality for the time optimal problem for nonlinear monotone parabolic equations are presented. For some problems governed by a parabolic variational inequality, these conditions are applied to a penalty problem associated with the inequality. Optimality conditions for the time optimal problem for the variational inequality are derived by passing to the limit in the penalizing term.

In this paper we are interested in a nonlinear parabolic evolution equation occurring in rheology. We give a probabilistic interpretation to this equation by associating a nonlinear martingale problem with it. We prove the existence of a unique solution, P, to this martingale problem. For any t, the time marginal of P at time t admits a density ρ(t,x) with respect to the Lebesgue measure, where the function ρ is the unique weak solution to the evolution equation in a well-chosen energy space. Next we introduce a simulable system of n interacting particles and prove that the empirical measure of this system converges to P as n tends to ∞. This propagation-of-chaos result ensures that the solution to the equation of interest can be approximated using a Monte Carlo method. Finally, we illustrate the convergence in some numerical experiments.

Domain Perturbation for Linear and Nonlinear Parabolic Equations

The initial-boundary value problems for parabolic equations with variable exponents of nonlinearity and time depended delay are considered. Existence and uniqueness of solutions of these problems are proved.

The Hardy inequality and nonlinear parabolic equations on Carnot groups

The problem related to controlled potential experiments in electrochemistry is studied. Ion transport is regarded as the superposition of diffusion and migration. Modelling of the experiment leads to a problem for a nonlinear parabolic equation with additional condition. Driven by the needs of theoretical analysis, from the point of view a inverse coefficient problem, we analyze the monotonicity of input-output mappings in inverse coefficient and source problems for this parabolic equation. Additionally, we extend the nonlinear parabolic equation to a more general case. Under some proper conditions, we investigate the existence of quasisolution of the generalized nonlinear parabolic equation.

Chapter 6. Energy methods for the investigation of localized regimes with peaking for parabolic second-order equations was published in Singular Solutions of Nonlinear Elliptic and Parabolic Equations on page 283.

ELENA BERETTA, MICHIEL BERTSCH & ROBERTA DAL PASSO Communicated by M. GUaTIN O. Introduction In this paper we study nonnegative solutions of the fourth-order equation (0.1) ut + (u"uxxx)x = O, where n is a real positive constant. Equation (0.1) is a nonlinear parabolic partial differential equation which is degenerate at points at which u vanishes. It arises in several applications (see the references in [2, 4, 5, 6]) in which u(x, t) stands for a nonnegative quantity. For example, in the study of the dynamics of thin liquid films u denotes the thickness of the film or another relevant (nonnegative) quantity. The case n = 1 models the flow in a Hele-Shaw cell, while n = 3 corresponds to viscous flow on a solid surface without slip driven by surface tension (see also [6, 7, 8] for a related equation). It is well known that fourth-order equations do not satisfy a maximum principle. In particular, classical solutions of the linear equation ut + u .... ---- 0 may become negative in finite time. BERNIS 8~; FRIEDMAN [-2] have observed that if n is positive, equation (0.1) has (generalized) solutions which remain nonnegative for all times. In addition, they show that the qualitative positivity of these solutions strongly depends on the value of n. In more recent papers by KADANOFF et al., a combination of numerical calculations and matched asymptotics [4] and the analysis of travelling-wave solutions [-5] leads to results which suggest a very rich structure of the solutions of (0.1). It is the purpose of the present paper to clarify several aspects of the nonlinear equation (0.1) and its dependence on the parameter n. Our results lead to several conjectures and open problems. As was pointed out in [4], for sufficiently small values of n, solutions of (0.1) which are initially strictly positive may vanish at some point x 0 after a finite time to, but, to the best of our knowledge, there is no rigorous proof of this phenomenon in the literature. On the other hand, it is of particular interest for applications; for example, if u denotes the thickness of a liquid film, it implies that at time to the film breaks at xo.

In this paper, the weighted average-based differential quadrature method is presented for solving one-dimensional homogeneous first-order non-linear parabolic partial differential equation. First, the given solution domain is discretized by using uniform discretization grid point. Next, by using Taylor series expansion we obtain central finite difference discretization of the partial differential equation involving with temporal variable associated with weighted average of partial derivative concerning spatial variable. From this, we obtain the system of nonlinear ordinary differential equations and it is linearized by using the quasilinearization method. Then by using the polynomial-based differential quadrature method for approximating derivative involving with spatial variable at specified grid point, we obtain the system of linear equation. Then they obtained linear system equation is solved by using the LU matrix decomposition method. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm and maximum absolute error norm and the local behavior of the solution is captured exactly. Numerical versus exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root mean square error norm and maximum absolute error norm presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving the non-linear parabolic equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution.