In a Banach space $X$ we consider the partial differential equation $$ (*)\quad D_{t}u(t,x)+(-1)^ma(x)D_{x}^{2m}u(t,x)-A(x)u(t,x)=f(t,x) $$ where $m$ is a positive integer, related to the rectangle $(0,T)\times(0,L)$ and the family of closed linear operators $\{A(x)\}_{x\in[0,L]}$. Under suitable assumptions we uniquely solve certain initial and boundary-value problems associated with $(*)$. Some applications are given when,for each $x,$ $A(x)$ is an explicit linear uniformly elliptic differential operator.

In this work, we present a new result ofexistence, uniqueness and maximal regularity for the restriction solutions of the bilaplacian transmission problem set in the juxtaposition of two rectangular bodies. The study is performed in the space $L^{p}( (-1, 0) \cup(0, \delta) ;X),$ $1 < p < \infty,$ where $\delta $ is a small parameter which is destined to tend to zero and $X$ is a UMD Banach space. The geometry of the bodies allows us to find an explicit representation of the solutions in virtue of the operational Dunford calculus. We then use essentially the famous Dore-Venni theorem among others for the analysis of the solutions.

We study the well posedness of the initial-value problem for a coupled semilinear reaction-diffusion system in Marcinkiewicz spaces $L^{(p_{1}, \infty)}(\Omega)\times L^{(p_{2}, \infty)}(\Omega)$. The exponents $p_{1},p_{2}$ of the initial-value space are chosen to allow the existence of self-similar solutions (when $\Omega=\mathbb{R}^{n}$). As a nontrivial consequence of our coupling-term estimates, we prove the uniqueness of solutions in the scaling invariant class $C([0,\infty);L^{p_{1}}(\Omega)\times L^{p_{2}}(\Omega))$ regardless of their size and sign. We also analyze the asymptotic stability of the solutions, show the existence of a basin of attraction for each self-similar solution and that solutions in $L^{p_{1} }\times L^{p_{2}}$ present a simple long-time behavior.

We investigate the KP-Burgers flow in the background of the shock profile solution of the KdV-Burgers equation. We show that the KdV-Burgers shock profile is stable and asymptotically stable under the KPB-I flow with respect to small perturbations in some spaces close to the energy space.

We prove that the parabolic equation $$f_t=\Delta f+F(x,f,\nabla f,t), $$ in $(\mathbb R^m\setminus\{0\})\times(0,T)$, $m\ge 3$, has removable singularities at $\{0\}\times (0,T)$ if $\|f\|_{L^{\infty}(\mathbb{R}^m\setminus\{0\}\times (0,T))} <\infty$ and $\|\nabla f\|_{L^{\infty}(\mathbb{R}^m\setminus\{0\}\times (0,T))} <\infty$. We also prove that the solution $u$ of the heat equation in $(\Omega\setminus\{0\})\times (0,T)$ has removable singularities at $\{0\}\times (0,T)$, $\Omega\subset\mathbb{R}^m$, $m\ge 3$, if and only if for any $0 < t_1 < t_2 < T$ and $\delta\in (0,1)$ there exists $\overline{B_{R_0}(0)}\subset\Omega$ depending on $t_1$, $t_2$ and $\delta$, such that $|u(x,t)|\le\delta |x|^{2-m}$ for any $0 <|x|\le R_0$ and $t_1\le t\le t_2$.

In this paper, we are interested in the Landau-de Gennes functional introduced to study the transition between the smectic and nematic phases of a liquid crystal. We define a reduced functional by constraining the director field to satisfy a non-homogeneous Dirichlet condition and we prove that, below a critical temperature and if some elastic coefficients are explicitly large, the minimizers have to be nematic phases.

In this paper we study the existence and multiplicity of solutions for the semilinear elliptic equation $-\Delta u = Q(x)f'(u)$ in an exterior domain with Neumann boundary conditions. We prove the existence of a positive ground state as well as a sign-changing solution under a double power growth condition on the nonlinearity.