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Cauchy problem for the complex Ginzburg-Landau type Equation with L p -initial data

Daisuke ShimotsumaTomomi YokotaKentarou Yoshii — 2014

Mathematica Bohemica

This paper gives the local existence of mild solutions to the Cauchy problem for the complex Ginzburg-Landau type equation u t - ( λ + i α ) Δ u + ( κ + i β ) | u | q - 1 u - γ u = 0 in N × ( 0 , ) with L p -initial data u 0 in the subcritical case ( 1 q < 1 + 2 p / N ), where u is a complex-valued unknown function, α , β , γ , κ , λ > 0 , p > 1 , i = - 1 and N . The proof is based on the L p - L q estimates of the linear semigroup { exp ( t ( λ + i α ) Δ ) } and usual fixed-point argument.

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