Convergence of quasilinear parabolic equations to semilinear equations

Quasilinear Parabolic Functional Evolution Equations

Based on our recent work on quasilinear parabolic evolution equations and maximal regularity we prove a general result for quasilinear evolution equations with memory. It is then applied to the study of quasilinear parabolic differential equations in weak settings. We prove that they generate Lipschitz semiflows on natural history spaces. The new feature is that delays can occur in the highest ...

Stability of Solutions of Quasilinear Parabolic Equations

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...

Optimal Control of Nonsmooth, Semilinear Parabolic Equations

This paper is concerned with an optimal control problem governed by a semilinear, nonsmooth operator differential equation. The nonlinearity is locally Lipschitz-continuous and directionally differentiable, but not Gâteaux-differentiable. Two types of necessary optimality conditions are derived, the first one by means of regularization, the second one by using the directional differentiability ...

Heteroclinic orbits of semilinear parabolic equations

Dimension splitting for quasilinear parabolic equations

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...