Quasilinear Evolution Equations and Parabolic Systems

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 ...

On Quasilinear Parabolic Evolution Equations in Weighted Lp-Spaces II

Our study of abstract quasi-linear parabolic problems in time-weighted Lp-spaces, begun in [17], is extended in this paper to include singular lower order terms, while keeping low initial regularity. The results are applied to reactiondiffusion problems, including Maxwell-Stefan diffusion, and to geometric evolution equations like the surface-diffusion flow or the Willmore flow. The method pres...

Parabolic systems and evolution equations on networks

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...

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...