In the current paper, we derive a rigorous convergence analysis for a broad range of splitting schemes applied to abstract nonlinear evolution equations, including the Lie and Peaceman–Rachford splittings. The analysis is in particular applicable to (possibly degenerate) quasilinear parabolic problems and their dimension splittings. The abstract framework is based on the theory of maximal dissi...

We use maximal Lp regularity to study quasilinear parabolic evolution equations. In contrast to all previous work we only assume that the nonlinearities are defined on the space in which the solution is sought for. It is shown that there exists a unique maximal solution depending continuously on all data, and criteria for global existence are given as well. These general results possess numerou...

Abstract. We bound the difference between solutions u and v of ut = a∆u+ divx f + h and vt = b∆v + divx g + k with initial data φ and ψ, respectively, by ‖u(t, ·)− v(t, ·)‖Lp(E) ≤ AE(t)‖φ−ψ‖ 2ρp L∞(Rn) +B(t)(‖a− b‖∞ + ‖∇x · f − ∇x · g‖∞ + ‖fu − gu‖∞ + ‖h− k‖∞)p |E| ηp . Here all functions a, f , and h are smooth and bounded, and may depend on u, x ∈ R, and t. The functions a and h may in additi...

and Applied Analysis 3 Mu et al. 19 studied the secondary critical exponent for the following p-Laplacian equation with slow decay initial values: ut div ( |∇u|p−2∇u ) u, x, t ∈ R × 0, T , u x, 0 u0 x , x ∈ R, 1.6 where p > 2, q > 1, and showed that, for q > q∗ c p − 1 p/N , there exists a secondary critical exponent ac p/ q 1 − p such that the solution u x, t of 1.6 blows up in finite time for...

In this paper we study solutions of the quasi-linear parabolic equations ∂u/∂t −∆pu = a(x)|u|q−1u in (0, T ) × Ω with Robin boundary condition ∂u/∂ν|∇u|p−2 = b(x)|u|r−1u in (0, T ) × ∂Ω where Ω is a regular bounded domain in IRN , N ≥ 3, q > 1, r > 1 and p ≥ 2. Some sufficient conditions on a and b are obtained for those solutions to be bounded or blowing up at a finite time. Next we give the a...

This paper is concerned with the state-constrained optimal control of the threedimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness and continuity for the state system are derived by employing maximal parabo...

This paper aims at solving an optimal control problem governed by anisotropic Allen-Cahn equation numerically. Therefor we first prove the Frechet differentiability of in time discretized parabolic under certain assumptions on involved quasilinearity and formulate order necessary conditions. As a next step, since anisotropies are general not smooth enough, convergence behavior controls is studi...