Second-Order Regularity for Parabolic p-Laplace Problems

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.

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.

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.

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.

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

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.

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

We study boundary-value problems with Dirichlet conditions for nonlinear parabolic equations with variable delay (i.e., delay is a function of time) and degeneration at the initial time. The existence and uniqueness of the classical solution of this problem are proved. The a priori estimates of this solution are obtained.

Cauchy problem for nonlinear parabolic equations with a gradient term

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.

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.

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

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

ALTERNATING-DIRECTION GALERKIN METHODS ON RECTANGLES