Expanding solutions of quasilinear parabolic equations

Nikolaos Roidos

Published 2017-12-25Version 1

We decompose locally in time maximal $L^{q}$-regular solutions of abstract quasilinear parabolic equations as a sum of a smooth term and an arbitrary small$-$with respect to the maximal $L^{q}$-regularity space norm$-$remainder. In view of this observation, we next consider the porous medium equation and the Swift-Hohenberg equation on manifolds with conical singularities. We write locally in time each solution as a sum of three terms, namely a term that near the singularity is expressed as a linear combination of complex powers and logarithmic integer powers of the singular variable, a term that decays to zero close to the singularity faster than each of the non-constant summands of the previous term and a remainder that can be chosen arbitrary small with respect e.g. to the $C^{0}$-norm. The powers in the first term are time independent and determined explicitly by the local geometry around the singularity, e.g. by the spectrum of the boundary Laplacian in the situation of straight conical tips. The case of the above two problems on closed manifolds is also considered and local space asymptotics for the solutions are provided.