Electronic journal of qualitative theory of differential equations · 2016 · 65 citations · 0 references
Natural SciencesMathematicsComputational Fluid DynamicsDifferential EquationsTransport PhenomenaBoundary ConditionsBoundary-value ProblemsInitial ConditionsInitial Value ProblemsPartial Differential EquationsBoundary Value Problems In TransportInitial Value Problem FormulationInitial-boundary Value ProblemsNonlinear Diffusion EquationsParabolic Boundary-value ProblemsStrong SolutionsWeak Solutions
In this paper, we study the initial boundary value problem of space-fractional diffusion equations. First, we introduce a family of potential wells. Then we show the existence of global weak solutions, provided that the initial energy $J(u_0)$ is positive and less than the potential well depth $d$. Finally, we establish the vacuum isolating and blow up of strong solutions.