A generalized model for acoustic boundary conditions to non-locally reacting boundaries is studied. We prove the existence, uniqueness, and asymptotic stability of global solutions to the mixed problem for the wave equation of Carrier type with acoustic boundary conditions for non-locally reacting boundaries. Additionally a nonlinear impenetrability condition is considered.

We consider the regularity estimate for the solution of the inhomogeneous heat equation in $(0,T)\times \Omega$ with 0-Dirichlet boundary conditions, where $ \Omega$ is a bounded domain in $\mathbb{R}^2$. We choose the external force $f$ from the Lorentz-Zygmund space which is not a reflexive Banach space, therefore our estimate cannot be obtained by a simple application of the estimate of the maximal regularity.

This paper studies the Cauchy problem for the MHD equations with mass diffusion in a bounded domain in $\mathbb{R}^{3}$. We use Tikhonov's fixed-point theorem to prove the existence and uniqueness of local solutions.

This paper is concerned with the existence of solutions for the quasilinear problem \[ \left \{ \begin{array}{lll} -div( \left|\nabla u\right| ^{N-2}\nabla u ) + \left| u\right|^{N-2}u= a(x)g(u)\quad \mbox{in}\ \Omega\\ \quad u=0\quad \mbox{on}\ \partial\Omega , \end{array} \right. \] where $\Omega \subset \mathbb{R}^N$ ($N\geq 2$) is an exterior domain; that is, $\Omega = \mathbb{R}^{N} \setminus \omega $, where $w \subset \mathbb{R}^{N}$ is a bounded domain, the nonlinearity $g(u)$ has an exponential critical growth at infinity and $a(x)$ is a continuous function and changes sign in $\Omega$. A variational method is applied to establish the existence of a nontrivial solution for the above problem.

In this paper we study the application of a general integral called a distributional Henstock-Kurzweil integral which includes the Henstock-Kurzweil integral and the wide Denjoy integral. We prove the existence of solutions of the wave equation involving the distributional Henstock-Kurzweil integral by using Schauder's fixed-point theorem.

In this paper we consider a Schrödinger equation on the circle with a quadratic nonlinearity. Thanks to an explicit computation of the first Picard iterate, we give a better description of the dynamic of the solution, whose existence was proved by C. E. Kenig, G. Ponce and L. Vega [15]. We also show that the equation is well posed in a space $\mathcal H^{s,p}(\mathbb T)$ which contains the Sobolev space $H^{s}(\mathbb T)$ when $p\geq 2$.

We prove nonexistence and uniqueness results of positive solutions for biharmonic supercritical equations $\Delta ^2 u = f(u)$ under Navier boundary conditions on a smooth bounded domain $\Omega \subset \mathbb{R}^N$. The results stand for suitable supercritical nonlinearities $f$ with some geometrical conditions on $ \Omega.$ We define the $h$-starlikeness of $\Omega$ and a classifying number $M(\Omega)$. This allows us to define a generalized critical exponent for these domains which play the role of the classical exponent $\frac{N+4}{N-4}$. Our approach is based on Rellich-Pohozaev type estimates. In particular, we construct some $h $-starlike domains which are topologically nontrivial where our results can apply.

We study the first eigenvalue of a $\lambda$-dependent cooperative elliptic system involving two quasilinear operators. By variational arguments, we find an expression for the limit of this eigenvalue as $\lambda \rightarrow -\infty$, which improves and extends (for gradient quasilinear systems) a result proved by Álvarez Caudevilla-López Gómez [4] and Dancer [9]. We apply this result to deduce the existence of strictly principal eigenvalues (i.e., whose eigenfunctions have both components positive) of a weighted system and extend the results proved in Cuesta-Ramos Quoirin [7] for the scalar case.

We consider a linear operator $L_m$ with variable coefficients of order $2m$ and we study the regularity of the very weak solution $u$ integrable in a bounded open smooth set $\Omega$, \[ \int_\Omega uL^*_m {\varphi} \,{\rm dx} = \int_\Omega {\varphi} d\mu\quad\forall\,{\varphi}\in C^{2m}( \overline \Omega) \] with $ \frac{{{\partial}}^j{\varphi}}{{{\partial}}\nu^j}=0$ on the boundary ${{\partial}}\Omega$ for $j{\leqslant} m-1$, where $L^*_m$ is the adjoint operator of $L_m$ and $\mu$ is in the space of weighted bounded Radon measures $M^1(\Omega,dist(x,{{\partial}}\Omega)^m)$. In particular, we show that the solution $u$ and all its derivatives of order $|{\gamma}|,\ |{\gamma}|{\leqslant} m-1,$ are in Lorentz spaces. If the measure on the right-hand side belongs to a smaller space such as $$ M^1(\Omega, dist(x,{{\partial}}\Omega)^{m-1+a}), \quad 0{\leqslant} a<1, $$ then all its derivatives o f order $|{\gamma}|,\ | {\gamma}|{\leqslant} m,$ are in Lorentz spaces.

We study solutions of the system \begin{eqnarray} \nonumber %26 \Delta u + k f(v)=h_1(x), \ x \in \Omega, \ u=0 \ \mbox{for $x \in \partial \Omega$} \\ \nonumber %26 \Delta v+kg(u)=h_2(x), \ x \in \Omega, \ v=0 \ \mbox{for $x \in \partial \Omega$} \nonumber \end{eqnarray} on a bounded smooth domain $\Omega \subset R^n$, with given functions $f(t)$, $g(t) \in C^2(R)$, and $h_1(x)$, $h_2(x) \in L^2(\Omega)$. When the parameter $k=0$, the problem is linear, and uniquely solvable. We continue the solutions in $k$ on curves of equiharmonic solutions. We show that in the absence of resonance the problem is solvable for any $h_1(x)$, $h_2(x) \in L^2(\Omega) $, while in the case of resonance we develop necessary and sufficient conditions for existence of solutions of E.M. Landesman and A.C. Lazer [12] type, and sufficient conditions for existence of solutions of D.G. de Figueiredo and W.-M. Ni [7] type. Our approach is constructive, and computationally efficient.

We consider the initial-boundary-value problem for the intermediate long-wave (ILW) equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary-value problem and the asymptotic behavior of solutions for large time.