We study the behavior of solutions of the Cauchy problem for a semilinear parabolic equation with a singular absorption term. We discuss the convergence of solutions to a singular stationary solution from above as time goes to infinity, and show that in a supercritical case a sharp estimate of the quenching rate can be determined explicitly when a specific growth rate of initial data is given. We also obtain a universal lower bound of the quenching rate which implies the optimality of the results. Proofs are given by a comparison method that is based on matched asymptotic expansion. We first determine a quenching rate of solutions by a formal analysis. Based on the formal analysis, we give a rigorous proof by constructing appropriate super- and subsolutions with the desired quenching rate.

In this paper we consider a model introduced in [3] which describes the interaction between the matter and the electromagnetic field from a unitarian standpoint. This model is based on a semilinear perturbation of the Maxwell equations and, in the magnetostatic case, reduces to the following nonlinear elliptic degenerate equation: $$\nabla \times (\nabla \times {\bf A})=W'(|{\bf A}|^2){\bf A},$$ where "$\nabla\times$" is the curl operator, $W:\mathbb R\to\mathbb R$ is a suitable nonlinear term, and ${\bf A}:\mathbb R^3\to\mathbb R^3$ is the gauge potential associated with the magnetic field ${\bf H}$. We prove the existence of a nontrivial finite energy solution with a kind of cylindrical symmetry. The proof is carried out by using a suitable variational framework based on the Hodge decomposition, which is crucial in order to handle the strong degeneracy of the equation. Moreover, the use of a natural constraint and a concentration-compactness argument are also required.

This paper is concerned with one-dimensional quadratic semilinear fourth-order Schrödinger equations. Motivated by the quadratic Schrödinger equations in the pioneering work of Kenig-Ponce-Vega [12], three bilinearities $uv,$ $\overline{uv},$ and $\overline{u}v $ for functions $u,v: \mathbb{R}\times[0,T]\mapsto \mathbb{C} $ are sharply estimated in function spaces $X_{s,b}$ associated to the fourth-order Schrödinger operator $i\partial_t+\Delta^2-\varepsilon\Delta$. These bilinear estimates imply local wellposedness results for fourth-order Schrödinger equations with quadratic nonlinearity. To establish these bilinear estimates, we derive a fundamental estimate on dyadic blocks for the fourth-order Schrödinger from the $[k,Z]$-multiplier norm argument of Tao [20].

We study the limit when $k\to\infty$ of the solutions of $ \partial_tu-\Delta u+f(u)=0$ in $\mathbb R^N\times (0,\infty)$ with initial data $k\delta$, when $f$ is a positive superlinear increasing function. We prove that there exist essentially three types of possible behaviour according to whether $f^{-1}$ and $F^{-1/2}$ belong or not to $L^1(1,\infty)$, where $F(t)=\int_0^t f(s)ds$. We use these results for providing a new and more general construction of the initial trace and some uniqueness and nonuniqueness results for solutions with unbounded initial data.

We establish local well-posedness results in weak periodic function spaces for the Cauchy problem of the Benney system. The Sobolev space $H^{1/2}\times L^2$ is the lowest regularity attained and also we cover the energy space $H^{1}\times L^2$, where global well posedness follows from the conservation laws of the system. Moreover, we show the existence of a smooth explicit family of periodic travelling waves of dnoidal type and we prove, under certain conditions, that this family is orbitally stable in the energy space.

A new boundary observability inequality for the Maxwell equations and the elastodynamic system is obtained. We use modified multipliers to obtain such an inequality as long as a geometric condition on the region holds and important parameters of the model are (numerically) related. This allow us to use the HUM to conclude a "simultaneous'' boundary exact controllability result.

In this paper we give a unified approach to the study of the convergence of a general family of linear sampling type operators which includes several operators very useful in signal reconstruction. We study both the pointwise and uniform convergence and the modular one in the general setting of Orlicz spaces. This creates the possibility of covering several settings such as $L^p$-spaces, the interpolation spaces, the exponential spaces and many others.