This work is devoted to the study of a nonlocal Ginzberg-Landau equation by the semigroup method and a nonlocal Fokker-Plank equation by the viscosity vanishing method. For the nonlocal Ginzberg-Landau equation, there exists a unique global solution in the set $C^0(\mathbb{ R}^+,\,H_0^{ \frac{ \alpha}{ 2}}(D))\cap L_{ loc}(\mathbb{ R}^+,\,H_0^{ \alpha}(D))$, for $\alpha\in (0,\,2)$. For the nonlocal Fokker-Plank equation, the regularity of the solution is weaker than that of the nonlocal Ginzberg-Landau equation due to the drift term.
Categories: math.AP