We show that, for some family of weights $\omega ,$ there are corresponding weighted Sobolev spaces $W_{\omega }^{m,p}$ on $ \mathbb {R}^{N}$ such that whenever $P(x,\partial)$ is a differential operator with $L^{\infty }$ coefficients and $P(x,\partial):W^{m,p}\rightarrow L^{p}$ is Fredholm for some $p\in (1,\infty),$ then $P(x,\partial):W_{\omega }^{m,p}\rightarrow L_{\omega }^{p}$ ($=W_{\omega }^{0,p}$) remains Fredholm with the same index. We also show that many spectral properties of $P(x,\partial)$ are closely related, or even the same, in the non-weighted and the weighted settings. The weights $\omega $ arise naturally from a feature of independent interest of the Fredholm differential operators in classical Sobolev spaces (``full'' decay transference), proved in the preparatory Section 2. A main virtue of the spaces $W_{\omega }^{m,p}$ is that they are well suited to handle nonlinearities that may be ill-defined or ill-behaved in non-weighted spaces. Together with the invariance results of this paper, this has proved to be instrumental in resolving various bifurcation issues in nonlinear elliptic PDEs.

We study the existence of solutions having singular limits for some four-dimensional semilinear elliptic problems with exponential nonlinearity and a singular source term given by Dirac masses with Navier boundary condition. In particular we extend the result of [2].

The identifiability of spatial variable coefficients for the vibrating string and Euler-Bernoulli beam are considered. It is shown that the coefficients can be determined by means of boundary control and observation in a finite time duration. These results can be considered as the generalization of infinite-time coefficients identifiability through the application of the Ingham-Beurling theorem.

In this paper, we are interested in nonlinear stochastic partial differential equations. Stochastic perturbations of a class of degenerate parabolic problems with homogeneous Dirichlet boundary conditions are considered. A time-discretization is used to prove the existence of a solution. The pivot-space method leads to the uniqueness. Then, applications to the porous media and the Buckley-Leverett equations are proposed.

We give a complete solution to the inverse scattering problem for the energy dependent Schrödinger equation on the full line when there are no bound states. This solution characterizes the image of the energy dependent scattering transform, generalizes the Deift-Trubowitz theorem for the standard Schrödinger equation in the absence of bound states, and indicates clearly the inverse transform by a simple inversion formula.

We discuss the global behavior of the weak solution of the Keller-Segel system of degenerate type. Asymptotic stability of the Barenblatt-Pattle solution and its convergence rate for the decaying weak solution in $L^1({\mathbb R}^n)$ is shown for the degenerated case $1 <{\alpha} < 2-\frac{2}{n}$. The method is based on the techniques applied to the Fokker-Plank equation due to Carrillo-Toscani [8] deriving from the explicit time decay of the free energy functional and some new estimates for the nonlinear interaction involving the critical type Sobolev inequality. We give the rigorous justification of those procedures via some approximating procedures.

In this paper we consider integral operators on the real line and derive certain sufficient conditions under which the operators act as bounded linear operators between the spaces of Stepanov bounded functions. Next, we find conditions that insure the operators act between spaces of Stepanov almost periodic, or between spaces of Stepanov almost automorphic, functions. We apply these results to ordinary differential equations and obtain the existence and uniqueness of bounded, almost periodic, and almost automorphic solutions.

We are interested in the initial-value problem associated to the Hirota-Satsuma equation in the real line $$ u_t+u_x-2uu_t+2u_x\int_x^{\infty}u_tdx'-u_{txx}=0, \quad x \in \mathbb R, $$ where $u$ is a real-valued function. This equation models the unidirectional propagation of shallow water waves as the well-known Korteweg-de Vries and Benjamin-Bona-Mahony equations. Here we show local well posedness for initial data in the space $$ \Omega_s=\{\phi \in H^s(\mathbb R) : -1 \notin \sigma(-\partial_x^2-2\phi)\} \ \text{if}\ s>\tfrac12, $$ and small initial data in $H^s(\mathbb R)$ if $0\le s \le \frac12$. We also prove global well posedness for small energy data in $H^1(\mathbb R)$.