Boundary-Value Problems for Nonlinear Parabolic Equations with Delay and Degeneration at the Initial Time

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.

Blow-up problems for nonlinear parabolic equations on locally finite graphs

Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems and Strichartz type estimates in mixed Lebesgue spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering.

Cauchy problem for nonlinear parabolic equations with a gradient term

Abstract. We study the inverse problem for nonlinear parabolic equation with variable thermal conductivity a ( x , t ) ${a(x,t)}$ which is a ill-posed problem. We regularize this problem by using the truncated integral method and then obtain the error estimate between the exact solution and the regularized solution. A numerical experiment is given for illustrating our method.

An optimal control problem for systems described by Fourier problem for nonlinear parabolic equations is studied. Control functions occur in the coefficients of the state equations. The existence of the optimal control in the case of final observation is proved.

1 Mean-square approximation for stochastic differential equations.- 2 Weak approximation for stochastic differential equations.- 3 Numerical methods for SDEs with small noise.- 4 Stochastic Hamiltonian systems and Langevin-type equations.- 5 Simulation of space and space-time bounded diffusions.- 6 Random walks for linear boundary value problems.- 7 Probabilistic approach to numerical solution of the Cauchy problem for nonlinear parabolic equations.- 8 Numerical solution of the nonlinear Dirichlet and Neumann problems based on the probabilistic approach.- 9 Application of stochastic numerics to models with stochastic resonance and to Brownian ratchets.- A Appendix: Practical guidance to implementation of the stochastic numerical methods.- A.1 Mean-square methods.- A.2 Weak methods and the Monte Carlo technique.- A.3 Algorithms for bounded diffusions.- A.4 Random walks for linear boundary value problems.- A.5 Nonlinear PDEs.- A.6 Miscellaneous.- References.

The method of lines in nonlinear boundary value problems for nonlinear parabolic equations

We study the extinction property of solutions to the Cauchy–Dirichlet problem for nonlinear parabolic equations of the order 2m with absorption potential in a semibounded cylinder (0,+∞) × Ω, where Ω is a bounded domain in ℝN, N ≥ 1. The sufficient conditions ensuring the extinction of a solution in a finite time, which depend on N,m, and q (where q is a parameter of the homogeneous nonlinearity in the main part of the equation), are obtained.

Optimal second-order regularity in the space variables is established for solutions to Cauchy–Dirichlet problems for nonlinear parabolic equations and systems of p-Laplacian type, with square-integrable right-hand sides and initial data in a Sobolev space. As a consequence, generalized solutions are shown to be strong solutions. Minimal regularity on the boundary of the domain is required, though the results are new even for smooth domains. In particular, they hold in arbitrary bounded convex domains.

On the well-posedness of a Fourier problem for nonlinear parabolic equations of higher order with variable exponents of nonlinearity

Some notions related to approximate solutions and to the approximation of extremum problems for nonlinear infinite-dimensional systems are proposed. Optimization problems for nonlinear parabolic equations with a fixed terminal state and on an infinite time interval, as well as for singular stationary systems with phase constraints, are illustrated by several examples.

Coefficient inverse problems are reformulated to a unified integral differential equation. The presented method of conversion of the considered inverse problems to a unified Volterra integral-differential equation gives an opportunity to distribute the acquired results also to analogous inverse problems for non-linear parabolic equations of different types.

On the uniqueness of the solution of a problem for non-linear parabolic equations without initial conditions

Let L be a linear second order parabolic partial differential operator, and suppose u is a solution to the differential inequality on satisfying u(x,0) = 0 for all . We show that the only solution for which is the trivial one provided the function h satisfies