Under suitable hypotheses on the nonlinear function $f$, the number of connected components of the complement of the nodal set of $\varphi$ is estimated when $\varphi$ is a solution of the elliptic equation $ -\Delta\varphi +f(\varphi) = 0$ in a bounded, open domain $\Omega$ with Dirichlet homogeneous boundary condition, and in the simplest case a dynamical consequence is derived for the corresponding semilinear heat equation. In addition, for simple domains such as a one-dimensional interval, a rectangle or a ball of arbitrary dimension, we establish the dynamical instability of solutions which do not have a constant sign in all the reasonable-looking cases.

We present in this paper an existence result as well a numerical method for a fluid-structure interaction problem. The structure motion is governed by the Vón Kármán equations and the fluid behavior by the viscous wave equation.

In this paper we study a class of nonlinear elliptic problems in $\mathbb R^N$, where the nonlinearity is of "jumping" type. The problems we study are perturbations of a linear problem at resonance. We use a perturbative method recently developed, and we get results which generalize to the case of an unbounded domain some of the theorems previously known for bounded domains.

In the framework of the potential well method, we consider the behavior of solutions to the problem (1.1)--(1.3) below with the critical Sobolev exponent. Roughly speaking, in the case where $\Omega$ is star-shaped, time-global solutions which intersect with the stable set at some time converge to zero uniformly for $x \in \Omega$ as $t \to +\infty$ and global solutions which intersect neither the stable nor the unstable sets blow up in infinite time in some sense and further have a property like a $\delta$-function in an appropriate sense as $t \to +\infty$ in the case when $\Omega$ is ball. Furthermore, for a kind of initial data the associated solution blows up at a finite time $T_{m}$, and its energy also satisfies: $J(u(t,\cdot)) = O(\log(T_{m}-t))$ as $t \uparrow T_{m}$.

This paper treats a two-dimensional, free-boundary problem arising in a mathematical model of chemical attack. A diffusion system is solved with a nonlinear condition on the free boundary, whose velocity is governed by the reaction of the concentrations of several compounds. An existence and uniqueness result of classical solutions is given in Hölder spaces, locally in time, for the corresponding Stefan-like problem to a system of parabolic equations with kinetic boundary condition.

In this paper a semilinear biharmonic problem involving critical growth is considered on a bounded convex domain. Rather complete results are obtained for the asymptotic behavior of positive solutions. The similar problem for the Laplacian was studied by Han, Rey and other authors.

We apply optimal control techniques to find approximate solutions to an inverse problem for the acoustic wave equation. The inverse problem (assumed to have a unique solution) is to determine the shape and reflection coefficient of a part of the boundary from partial measurements of the acoustic signal. The sought functions are treated as controls and the goal - quantified by an objective functional - is to drive the model solution close to the experimental data by adjusting these functions. The problem is solved by finding the optimal control pair, which minimizes the objective functional. Then by driving the "cost of the control" to zero one proves that the sequence of optimal controls converges to the solution of the inverse problem.

We consider a class of dynamic models for elastomers involving nonlinear viscoelastic (hysteresis) as well as nonlinear finite elastic components of the constitutive laws. Existence and uniqueness results are presented along with sample numerical fits to experimental data to demonstrate the efficacy of the models.

This paper is concerned with the existence of steady-state solutions for $$ \left\{ \begin{array}{ll} u_t = \epsilon^2 u_{xx} - (u-a(x))(u-b(x))(u-c(x))\quad & \mbox{in}~(0,1)\times(0,\infty),\\ u_x(0,t) = u_x(1,t) = 0\quad & \mbox{in}~(0,\infty). \end{array} \right. $$ Here $a, b$ and $c$ are $C^2$-functions satisfying $b = (a+c)/2$ and $c > a$. By using upper and lower solutions it is proved that there exist stable steady states with transition layers near any points where $c(x)-a(x)$ has its local minimum.

We consider time optimal control problems governed by semilinear parabolic equations with pointwise state constraints and unbounded controls. We derive a Pontryagin's principle for boundary controls. We prove a regularity result for the gradient of the state variable and by this way we are able to define a Hamiltonian functional which intervenes in an optimality condition satisfied by the optimal terminal time.

In this short note we investigate the asymptotic behavior of positive minimizing solutions $u_\epsilon$ to the equation $\Delta u_\epsilon = N(N-2) f(x) u_\epsilon^{p-\epsilon}$ in $\Omega$ and $u_\epsilon = 0$ on $\partial\Omega$, where $\Delta$ stands for the Euclidean Laplacian with the minus sign convention, $\Omega$ is a smooth bounded domain in ${\mathbb R}^N$, $p = (N+2)/(N-2)$ is the critical Sobolev exponent, and $f$ belongs to a fairly general class of functions.

We study the growth of the $H^s$-norm for solutions of the modified Korteweg-de Vries equation, corresponding to data in $H^s$ for noninteger values of $s$ in the case where global solutions exist. The presence of conservation laws and the local existence theory permit us to obtain upper "polynomial" bounds, for the $H^s$-norm of these solutions, with power depending on the distance to the closest integer to $s$.

Our aim is to prove, by a constructive process with strong convergence, the existence of a minimal solution for some quasilinear degenerate elliptic equations of the type $$-div \; A(x, \nabla u)= F(x,u) \; \mbox{in} \; \Omega .$$ Our proof uses rearrangement techniques and provides comparisons.

We study solutions of the parabolic equation $u_t = \Delta u + u^p$. We wish to extend some results of Giga and Kohn to the situations where the solution, $u$, is defined on a $C^{2,\alpha}$ domain and satisfies the Dirichlet or the Neumann boundary condition.

We study the solution for the initial boundary value problem of a nonlocal semilinear heat equation. It is well-known that the solution quenches in finite time for certain choices of initial data. We first prove that there is only one quenching point for symmetric initial data with one peak. Then we derive a quenching rate estimate. It turns out that the constant in the quenching rate estimate depends on the solution itself due to the nonlocal nonlinearity.

We consider bounded positive solutions $\alpha$ of rotationally symmetric harmonic map equations. We study the continuity of the maps $\alpha' (0) \mapsto \alpha (\infty)$ and $\alpha (1) \mapsto \alpha (\infty)$ in connection with the Dirichlet problem at infinity. Regularity at zero, local properties and conditions for positive solutions to be blowing up, unbounded, or bounded are discussed.

We prove that the surface-diffusion flow and the volume-preserving mean curvature flow can drive embedded hypersurfaces to self-intersections.