Nonlinear Differential Equations and Applications (NoDEA)最新文献

This paper is devoted to describing a linear diffusion problem involving fractional-in-time derivatives and self-adjoint integro-differential space operators posed in bounded domains. One main concern of our paper is to deal with singular boundary data which are typical of fractional diffusion operators in space, and the other one is the consideration of the fractional-in-time Caputo and Riemann–Liouville derivatives in a unified way. We first construct classical solutions of our problems using the spectral theory and discussing the corresponding fractional-in-time ordinary differential equations. We take advantage of the duality between these fractional-in-time derivatives to introduce the notion of weak-dual solution for weighted-integrable data. As the main result of the paper, we prove the well-posedness of the initial and boundary-value problems in this sense.

The necessity of a Maximum Principle arises naturally when one is interested in the study of qualitative properties of solutions to partial differential equations. In general, to ensure the validity of these kinds of principles one has to consider some additional assumptions on the ambient manifold or on the differential operator. The present work aims to address, using both of these approaches, the problem of proving Maximum Principles for second order, elliptic operators acting on unbounded Riemannian domains under Dirichlet boundary conditions. Hence there is a natural division of this article in two distinct and standalone sections.

We consider the initial-boundary value problem of the nonlinear Schrödinger equation on the half plane with a nonlinear Neumann boundary condition. After establishing the boundary Strichartz estimate in (L^2({mathbb {R}}^2_+)) and (H^s({mathbb {R}}^2_+)), we consider the time local well-posedness of the problem in (L^2({mathbb {R}}^2_+)) and (H^s({mathbb {R}}^2_+)).

We study a one-dimensional one-phase Stefan problem with a Neumann boundary condition on the fixed part of the boundary. We construct the unique self-similar solution, and show that starting from arbitrary initial data, solution orbits converge to the self-similar solution.

In this paper we prove existence and regularity results for a class of parabolic problems with irregular initial data and lower order terms singular with respect to the solution. We prove that, even if the initial datum is not bounded but only in (L^1(Omega )), there exists a solution that “instantly” becomes bounded. Moreover we study the behavior in time of these solutions showing that this class of problems admits global solutions which all have the same behavior in time independently of the size of the initial data.

In this work, we are interested in to study removability of a singular set in the boundary for some classes of quasilinear elliptic equations. We will approach this question in two different ways: through an asymptotic behavior at the infinity of the solutions, or through conditions in the Sobolev norm of solutions along the direction of the singular set.

On the two-sphere (Sigma ), we consider the problem of minimising among suitable immersions (f ,:Sigma rightarrow mathbb {R}^3) the weighted (L^infty ) norm of the mean curvature H, with weighting given by a prescribed ambient function (xi ), subject to a fixed surface area constraint. We show that, under a low-energy assumption which prevents topological issues from arising, solutions of this problem and also a more general set of “pseudo-minimiser” surfaces must satisfy a second-order PDE system obtained as the limit as (p rightarrow infty ) of the Euler–Lagrange equations for the approximating (L^p) problems. This system gives some information about the geometric behaviour of the surfaces, and in particular implies that their mean curvature takes on at most three values: (H in { pm Vert xi HVert _{L^infty } }) away from the nodal set of the PDE system, and (H = 0) on the nodal set (if it is non-empty).

We deal with the relaxed area functional in the strict BV-convergence of non-smooth maps defined in domains of generic dimension and taking values into the unit circle. In case of Sobolev maps, a complete explicit formula is obtained. Our proof is based on tools from Geometric Measure Theory and Cartesian currents. We then discuss the possible extension to the wider class of maps with bounded variation. Finally, we show a counterexample to the locality property in case of both dimension and codimension larger than two.

In this paper we study existence and uniqueness of solutions to Dirichlet problems as

where (Omega ) is an open bounded subset of ({{,mathrm{mathbb {R}},}}^N) ((Nge 2)) with Lipschitz boundary, (g:mathbb {R}rightarrow mathbb {R}) is a continuous function and f belongs to some Lebesgue space. In particular, under suitable saturation and sign assumptions, we explore the regularizing effect given by the absorption term g(u) in order to get solutions for data f merely belonging to (L^1(Omega )) and with no smallness assumptions on the norm. We also prove a sharp boundedness result for data in (L^{N}(Omega )) as well as uniqueness if g is increasing.

In this article, we study the ill-posedness of the viscous nonlinear wave equation for any polynomial nonlinearity in negative Sobolev spaces. In particular, we prove a norm inflation result above the scaling critical regularity in some cases. We also show failure of (C^k)-continuity, for k the power of the nonlinearity, up to some regularity threshold.