Attention is given to the question of well-posedness in Hadamard's classical sense for nonlinear evolution equations of the form \begin{equation} \frac{du}{dt} +L u = N (u), \qquad u(0) =\phi. \tag*{(0.1)} \end{equation} In view are various classes of nonlinear wave equations, nonlinear Schrödinger equations, and the (generalized) KdV equations. Equations of type (0.1) are often well posed in a scale $X_s$, say, of Banach spaces, at least for $s$ large enough. Here, increasing values of $s$ correspond to more regularity; thus $X_r\subset X_s $ if $r>s$. For smaller values of $s$, some equations of the form in (0.1) are well-posed in a conditional sense that the uniqueness aspect depends upon the imposition of auxiliary conditions. In the latter context, it is natural to inquire whether or not the auxiliary conditions are essential to securing uniqueness. It is shown here that for a conditionally well-posed Cauchy problem (0.1), the auxiliary specification is removable if a certain persistence of regularity holds. As a consequence, it will transpire that a conditionally well-posed problem (0.1) is (unconditionally) well-posed if the aforementioned persistence property holds. These results are applied to study several recent conditional well-posedness results for the KdV equation, nonlinear Schrödinger equations, and nonlinear wave equations. It is demonstrated that the auxiliary conditions used to secure the uniqueness are all removable and the corresponding Cauchy problems are, in fact, unconditionally well-posed as long as their classical solutions exist globally. In addition, the well-posedness for an initial-boundary-value problem for the KdV equation posed in a quarter plane is also considered. An affirmative answer is provided for a uniqueness question left open in a recent paper of Colliander and Kenig [14].

It has been known from the beginning of the 1980's that the global $L^1$ solutions of the classical porous medium equation with absorption $u_t = \Delta u^m - u^p$ in $\mathbb R^N \times \mathbb R_+$, with $m,p>1$ change their large-time behavior at the critical absorption exponent $p_0=m+2/N$. We show that, actually, there exists an infinite sequence $\{p_k, \, k \ge 0\}$ of critical exponents generating a countable subset of different non-self-similar asymptotic patterns. The results are extended to the fully nonlinear dual porous-medium equation with absorption.

We study a geometric free-boundary problem for a bicrystal in which a grain boundary is attached at a groove root to the exterior surface of the bicrystal. Mathematically, this geometric problem couples motion by mean curvature of the grain boundary with surface diffusion of the exterior surface. If the groove root effects are localized, it is realistic to look for traveling-wave solutions. We show that traveling-wave solutions can be determined via solutions of Problem $P_{\Psi}$ in which $$\Psi_{sss}= \sin \Psi, \quad s \in (-\infty, \, 0) \cup (0, \, \infty), \quad \Psi(\pm \infty)=0,$$ and appropriate jump conditions are prescribed at $s=0$. We prove existence of solutions to Problem $P_{\Psi}$ for all $m,$ $0 \le m <2$, where $m$ denotes the ratio of the surface energies of the grain boundary and of the exterior surface. We show numerically that for $\,\approx\! 1.81 < m \le 2$, the corresponding solutions to the original geometric problem are not single-valued as functions of $x$, where $x$ varies along the unperturbed exterior surface of the bicrystal. We refer to these solutions as "nonclassical traveling-wave solutions."

We prove local estimates for solutions of equation (1.1), which are applied to study existence and uniqueness of so-called large solutions, i.e., solutions which blow up uniformly at the boundary, as well as to the study of solutions in the whole space.

This paper is devoted to the the study of density-dependent, incompressible Navier-Stokes equations with periodic boundary conditions, or in the whole space. We aim at stating well-posedness in functional spaces as close as possible to the ones imposed by the scaling of the equations. Preliminary results have been obtained in [5] under the assumption that the density is close to a constant. Getting rid of this assumption (by allowing smoother data if necessary) is the main motivation of the present paper. Local well-posedness is stated for data $(\rho_0,u_0)$ such that $(\rho_0-cste)\in H^{{{{\frac N2}}}+\alpha}$ and $\inf\rho_0>0$, and $u_0\in H^{{{{\frac N2}}}-1+\beta}$. The indices $\alpha,\beta>0$ may be taken arbitrarily small. We further derive a blow-up criterion which entails global well-posedness in dimension $N=2$ if there is no vacuum initially.

In this paper we apply a set-up introduced by R. K. Miller to transform a linear, inhomogeneous Cauchy problem for the generator of a semigroup on a Banach space into a homogeneous one for a matrix operator, which is the generator of a semigroup on a suitable product space. By using restriction theorems and extrapolation spaces we obtain new results for the inhomogeneous Cauchy problem for Hille-Yosida operators in Favard spaces.

In this paper we consider the following nonlinear elliptic problem: $-\Delta u=Ku^3,$ $u>0$ in $\Omega$, $u=0$ on $\partial\Omega$, where $K$ is a positive function and $\Omega$ is a bounded domain of $R^4$. We prove a version of the Morse lemma at infinity for this problem, which allows us to describe the critical points at infinity of the associated variational problem. Using a topological argument, we are able to prove an existence result.

In this article, we are interested in the existence, uniqueness, and regularity of solutions of fully nonlinear parabolic equations, with initial data $u_0$, in the whole space $\mathbb R^n$. Our main result is the existence of a strictly subquadratic solution with a local $C^{1,\beta}$ regularity with respect to the space variable, assuming $C^{1,\alpha}$ regularity on $u_0$ and local uniform ellipticity of the equation. Our proof relies on a result of N. Zhu which shows the local $C^{1,\beta}$ regularity of the solution provided it is Lipschitz continuous and H\"older continuous in t, with an exponent $ {\gamma>\frac{1}{2}}$; we obtain this last property through a continuous-dependence result. Then we investigate further regularity for the solution using results of L. Wang.