This paper gives the local existence of mild solutions to the Cauchy problem for the complex Ginzburg-Landau type equation ∂u ∂t − (λ + iα)∆u + (κ + iβ)|u| q−1 u − γu = 0 in R N × (0, ∞) with L p -initial data u0 in the subcritical case (1 6 q < 1 + 2p/N), where u is a complex-valued unknown function, α, β, γ, κ ∈ R, λ > 0, p > 1, i = √ −1 and N ∈ N. The proof is based on the L p -L q estimates of the linear semigroup {exp(t(λ + iα)∆)} and usual fixed-point argument.
We consider the mixed problem for the hyperbolic partial differential-functional equation of the first order \[ D_xz(x,y) = f(x,y,z_{(x,y)}, D_yz(x,y)), \] where $z_{(x,y)} \: [-\tau ,0] \times [0,h] \rightarrow \mathbb{R}$ is a function defined by $z_{(x,y)}(t,s) = z(x+t, y+s)$, $(t,s) \in [-\tau ,0] \times [0,h]$. Using the method of bicharacteristics and the method of successive approximations for a certain integral-functional system we prove, under suitable assumptions, a theorem of the local existence of generalized solutions of this problem.