We show the global existence of solutions of a reaction-diffusion system with the nonlinear terms $|x|^{\sigma_j}u^{p_j}v^{q_j}$. The proof is based on the existence of supersolutions and the comparison principle. We also prove that uniqueness of the global solutions holds in the superlinear case by contraction argument. Our conditions for the global existence are optimal in view of the nonexistence results proved by Yamauchi [12].

We examine the question of the minimal Sobolev regularity required to construct local solutions to the Cauchy problem for the Benney--Luke (BL) and generalized Benney--Luke (gBL) equations. As a consequence we prove that the initial-value problems are globally well-posed in the energy space.

We study the initial-boundary value problems for the nonlinear nonlocal equation on a segment $\left( 0,a\right) $ \begin{equation} \left\{ \begin{array}{c} u_{t}+\lambda \left\vert u\right\vert \text{ }u+C_{1}\int_{0}^{x}\frac{ u_{ss}(s,t)}{\sqrt{x-s}}ds=0,\text{ }t>0, \\ u(x,0)=u_{0}(x), \\ u(a,t)=h_{1}(t),u_{x}(0,t)=h_{2}(t),t>0, \end{array} \right. \label{2} \end{equation} where $\lambda \in \mathbf{R}$ and the constant $C_{1}$ is chosen by the condition of the dissipation, such that $ {\rm Re\,}C_{1}p^{\frac{3}{2}}>0$ for ${\rm Re\,}p=0.$ The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem and to find the main term of the asymptotic representation of solutions.

We study the Dirichlet problem for the non-local diffusion equation $u_t=\int\{u(x+z,t)-u(x,t)\}{\,\mathrm{d}\mu}(z)$, where $\mu$ is a $L^1$ function and $``u=\varphi$ on $\partial\Omega\times(0,\infty)$'' has to be understood in a non-classical sense. We prove existence and uniqueness results of solutions in this setting. Moreover, we prove that our solutions coincide with those obtained through the standard ``vanishing viscosity method'', but show that a boundary layer occurs, the solution does not take the boundary data in the classical sense on $\partial\Omega$, a phenomenon related to the non-local character of the equation. Finally, we show that in a bounded domain, some regularization may occur, contrary to what happens in the whole space.

Let $\Omega \subset \mathbb{R}^{N}$ be a smooth bounded domain and let $m$ be a $T$-periodic function such that $m_{\mid \Omega \times \left( 0,T\right) }\in L^{r}\left( \Omega \times \left( 0,T\right) \right) $ for some $r>N+2$ and $\int_{0}^{T} \,{\rm esssup}\, _{x\in \Omega }m\left( x,t\right) dt>0.$ Let $\lambda _{1}\left( m\right) $ be the (unique) positive principal eigenvalue of the Dirichlet periodic parabolic problem $ Lu=\lambda mu$ in $\Omega \times \mathbb{R}$, $u=0$ on $\partial \Omega \times \mathbb{R},$ $u>0$ in $\Omega \times \mathbb{R}.$ We prove a formula for $\lambda _{1}\left( m\right) $ which is an analogous of the well known variational expression for principal eigenvalues of self-adjoint elliptic problems. As a direct consequence we obtain monotonicity results for $ \lambda _{1}\left( m\right) $ with respect to the domain $\Omega $ and with respect to the zero order coefficient of the differential operator $L$.

We analyze the positive solutions to the singular boundary value problem \[-\Delta u = \lambda[ f(u)-1/u^{\alpha}]; x \in \Omega\] \[u = 0; \, x \in \partial\Omega,\] where $f$ is a $C^2$ function in $(0,\infty)$, $f(0)\geq 0,f^{'}>0, \lim_{s\rightarrow\infty}\frac{f(s)}{s}=0, \lambda$ is a positive parameter, $\alpha \in (0,1)$ and $\Omega$ is a bounded region in $ R^{n}, n \geq 1 $ with $C^{2+\gamma}$ boundary for some $\gamma \in (0,1)$. In the case $n=1$ we use the quadrature method and for $n>1$ we use the method of sub-super solution to establish our results.