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Register a new userWell-posedness for weak and strong solutions of non-homogeneous initial boundary value problems for fractional diffusion equations
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We study the well-posedness for initial boundary value problems associated with time fractional diffusion equations with non-homogenous boundary and initial values. We consider both weak and strong solutions for the problems. For weak solutions, we introduce a new definition of solutions which allows to prove the existence of solution to the initial boundary value problems with non-zero initial and boundary values and non-homogeneous terms lying in some arbitrary negative-order Sobolev spaces. For strong solutions, we introduce an optimal compatibility condition and prove the existence of the solutions. We introduce also some sharp conditions guaranteeing the existence of solutions with more regularity in time and space.
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A reaction-diffusion equation with power nonlinearity formulated either on the half-line or on the finite interval with nonzero boundary conditions is shown to be locally well-posed in the sense of Hadamard for data in Sobolev spaces. The result is established via a contraction mapping argument, taking advantage of a novel approach that utilizes the formula produced by the unified transform method of Fokas for the forced linear heat equation to obtain linear estimates analogous to those previously derived for the nonlinear Schrodinger, Korteweg-de Vries and good Boussinesq equations. Thus, the present work extends the recently introduced unified transform method approach to well-posedness from dispersive equations to diffusive ones.
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The diffusion equation is a universal and standard textbook model for partial differential equations (PDEs). In this work, we revisit its solutions, seeking, in particular, self-similar profiles. This problem connects to the classical theory of special functions and, more specifically, to the Hermite as well as the Kummer hypergeometric functions. Reconstructing the solution of the original diffusion model from novel self-similar solutions of the associated self-similar PDE, we infer that the $t^{-1/2}$ decay law of the diffusion amplitude is {it not necessary}. In particular, it is possible to engineer setups of {it both} the Cauchy problem and the initial-boundary value problem in which the solution decays at a {it different rate}. Nevertheless, we observe that the $t^{-1/2}$ rate corresponds to the dominant decay mode among integrable initial data, i.e., ones corresponding to finite mass. Hence, unless the projection to such a mode is eliminated, generically this decay will be the slowest one observed. In initial-boundary value problems, an additional issue that arises is whether the boundary data are textit{consonant} with the initial data; namely, whether the boundary data agree at all times with the solution of the Cauchy problem associated with the same initial data, when this solution is evaluated at the boundary of the domain. In that case, the power law dictated by the solution of the Cauchy problem will be selected. On the other hand, in the non-consonant cases a decomposition of the problem into a self-similar and a non-self-similar one is seen to be beneficial in obtaining a systematic understanding of the resulting solution.
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