We introduce and study an initial and boundary-value problem for the Zakharov-Kuznetsov equation posed on an infinite strip of ${\mathbb{R}^{d+1}}$, $d=1,2$. After establishing a suitable trace theorem, we first consider the linearized case and define the corresponding semigroup on $L^2$ and prove that it has a global smoothing effect. Then we proceed to the nonlinear case and use the smoothing effect to prove in both dimensions the existence of (unique when $d=1$) global weak solutions of the initial and boundary problem with null boundary conditions and $L^2$ initial data.

We give a new approach to general Emden-Fowler equations and systems of the form \begin{equation*} (E_{\varepsilon })-\Delta _{p}u=-{\rm div}(\left\vert \nabla u\right\vert ^{p-2}\nabla u)=\varepsilon \left\vert x\right\vert ^{a}u^{Q}, \end{equation*} \begin{equation*} (G)\left\{ \begin{array}{c} -\Delta _{p}u=-{\rm div}(\left\vert \nabla u\right\vert ^{p-2}\nabla u)=\varepsilon _{1}\left\vert x\right\vert ^{a}u^{s}v^{\delta }, \\ -\Delta _{q}v=-{\rm div}(\left\vert \nabla v\right\vert ^{q-2}\nabla u)=\varepsilon _{2}\left\vert x\right\vert ^{b}u^{\mu }v^{m},% \end{array}% \right. \end{equation*}% where $p,q,Q,\delta, \mu, s,m,$ $a,b$ are real parameters, $p,q\neq 1,$ and $% \varepsilon, \varepsilon _{1},\varepsilon _{2}=\pm 1.$ In the radial case we reduce the problem $(G)$ to a quadratic system of four coupled first-order autonomous equations of Kolmogorov type. In the scalar case the two equations $(E_{\varepsilon })$ with source ($\varepsilon =1)$ or absorption (% $\varepsilon =-1)$ are reduced to a unique system of order 2. The reduction of system $(G)$ allows us to obtain new local and global existence or nonexistence results. We consider in particular the case $\varepsilon _{1}=\varepsilon _{2}=1.$ We describe the behaviour of the ground states when the system is variational. We give a result of existence of ground states for a nonvariational system with $p=q=2$ and $s=m>0,$ that improves the former ones. It is obtained by introducing a new type of energy function. In the nonradial case we solve a conjecture of nonexistence of ground states for the system with $p=q=2$, $\delta =m+1$ and $\mu =s+1.$

This paper proves the multiplicity of positive solutions for the following class of quasilinear problems: $$ \begin{cases} -\epsilon^{p}\Delta_{p}{u}+(\lambda A(x)+1)|u|^{p-2}u=f(u), \,\,\, \mathbb{R}^{N}\\ u(x)>0 \,\,\, \mbox{in} \,\, \mathbb{R}^{N}, \end{cases} $$ where $\Delta_{p}$ is the p-Laplacian operator, $ N >p \geq 2$, $\lambda$ and $\epsilon$ are positive parameters, $A$ is a nonnegative continuous function and $f$ is a continuous function with subcritical growth. Here, we use variational methods to get multiplicity of positive solutions involving the Lusternick-Schnirelman category of ${\rm{int}}A^{-1}(0)$ for all sufficiently large $\lambda$ and small~$\epsilon$.

The stationary Boltzmann equation for hard and soft forces in the context of a two-component gas is considered in the slab when the molecular masses of the two components are different. An $L^{1}$ existence theorem is proved when one component satisfies a given data profile and the other component satisfies diffuse reflection at the boundaries. Weak $L^{1}$ compactness is extracted from the control of the entropy production term of the mixture.

Motivated by diffusion processes on metric graphs and ramified spaces, we consider an abstract setting for interface problems with coupled dynamic boundary conditions belonging to a quite general class. Beside well posedness, we discuss positivity, $L^\infty$-contractivity and further invariance properties. We show that the parabolic problem with dynamic boundary conditions enjoys these properties if and only if its counterpart with time-independent boundary conditions does also. Furthermore, we prove continuous dependence of the solution to the parabolic problem on the boundary conditions in the considered class.

We prove optimal error estimates in energy norms and related norms for a space semidiscrete and for a fully discrete approximation of the Cahn-Hilliard-Gurtin equations with source terms. Numerical simulations in one and two space dimensions illustrate the theoretical results. We also prove convergence to equilibrium for the fully discrete scheme without source terms by the use of the Łojasiewicz inequality.