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Existence and smoothing effects of the initial-boundary value problem for ∂u/∂t−Δσ(u) = 0 in time-dependent domains

Nakao Mitsuhiro

Opuscula Mathematica

2023

Vol. 43, no. 5

703--734

We show the existence, smoothing effects and decay properties of solutions to the initial-boundary value problem for a generalized porous medium type parabolic equations of the form ut − Δσ(u) = 0 in Q(0, T) with the initial and boundary conditions u(0) = u0 and u(t)|∂Ω(t) = 0, where Ω(t) is a bounded domain in RN for each t ≥ 0 and [formula]. Our class of σ(u) includes σ(u) = |u|mu, σ(u) = u log(1 + |u|m), 0 ≤ m ≤ 2, and [formula]. We derive precise estimates for ∥u(t)∥Ω(t),∞ and ∥∇σ(u(t))∥2Ω(t),2, t > 0, depending on ∥u0∥Ω(0),r and the movement of ∂Ω(t).

Identification of a Quasilinear Parabolic Equation from Final Data

Fernandez L. A., Pola C.

International Journal of Applied Mathematics and Computer Science

2001

Vol. 11, no 4

859-879

We study the identification of the nonlinearities A, vec{b} and c appearing in the quasilinear parabolic equation y_t - div (A(y) nabla y + vec{b} (y)) + c(y) = u in Omega x (0,T), assuming that the solution of an associated boundary value problem is known at the terminal time, y(x,T), over a (probably small) subset of Omega, for each source term u. Our work can be divided into two parts. Firstly, the uniqueness of A, vec{b} and c is proved under appropriate assumptions. Secondly, we consider a finite-dimensional optimization problem that allows for the reconstruction of the nonlinearities. Some numerical results in the one-dimensional case are presented, even in the case of noisy data.

The finite difference scheme for the solution of one quasilinear equation

Ciegis R., Ciegis R., Rate P.

Demonstratio Mathematica

2001

Vol. 34, nr 3

723-732

In this paper we are interested in the solution of one-dimensional quasi-linear diffusion-reaction equation. The nonlinear reaction term includes the first derivative in space of the solution. We use the finite difference method to discretize this problem. The modification of a general methodology for investigation of difference schemes approximating non-stationary differential equations is used and the results for the stability and convergence of the numerical solution are proved. The convergence of the discrete derivative of the solution is proved in the maximum norm.