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Unbounded solutions to defocusing parabolic systems. (English) Zbl 1363.35147
Summary: We give existence theorems for the Cauchy problem of a large class of semi-linear parabolic equations in $$L^\infty$$, $$L^\infty\cap L^p$$ or $$L^\infty\cap \dot W^{1,p}$$ using a contracting map argument. We then construct integral solutions to parabolic equations with data growing at infinity and defocusing nonlinearity, and give an example of instantaneous blow up when the nonlinearity is focusing and the initial data has tame growth.

##### MSC:
 35K15 Initial value problems for second-order parabolic equations 35C15 Integral representations of solutions to PDEs 35Q56 Ginzburg-Landau equations
##### Keywords:
parabolic system; Cauchy problem; mild solution; blow-up