We consider the problem \begin{equation}\nonumber \left\{ \begin{array}{ll} -\Delta u=K(x)u^{p_{\epsilon}} & \hbox{ in }\mathbb R^n\\ u>0 & \hbox{ in }\mathbb R^n \end{array}\right. \end{equation} where $p=\frac{n+2}{n-2}$, $p_{\epsilon}=p-\epsilon$, $n\geq 3$, $\epsilon>0$ and $K(x)>0$ in $\mathbb R^n$. We prove an existence and multiplicity result for single peaked solutions of our problem concentrating at a fixed critical point of $K(x)$ and some other related results.

We consider the Cauchy problem for a parabolic-elliptic system of drift-diffusion type in $\mathbb R^2$, modeling chemotaxis and self-attracting particles, with $L^1$-initial data. Under the assumption that the total mass of nonnegative initial data is less than $8\pi$, by using similarity arguments, it is shown that the nonnegative solution converges to a radially symmetric self-similar solution at rate $o(t^{-1+1/p})$ in the $L^p$-norm $(1\le p\le\infty)$ as time goes to infinity.

In this paper we study the regularity of the so-called very weak solution for a parabolic equation. This unique solution is only integrable over the parabolic cylinder. The initial data and the right-hand side of the linear parabolic equation are functions integrable with respect to the weight function which corresponds to the distance function. In particular, we prove some global regularity of the space-gradient in Lorentz spaces. The regularity with respect to the time derivative is obtained under the condition that the linear operator is time independent and self-adjoint via $m$-accretive theory.

Of concern is the nonlinear parabolic problem with nonlinear dynamic boundary conditions \begin{align*} & u_t +\,{\rm div}(F(u))=\,{\rm div}({\mathcal A}\nabla u),\qquad u(0,x)=f(x), \\ & u_t +\beta{{{\partial}^{\mathcal A}_{\nu}}} u+\gamma(x, u)-q\beta {\Delta_{\rm LB}} u=0, \end{align*} for $x\in \Omega\subset \mathbb R^N$ and $t\ge0$; the last equation holds on the boundary ${\partial}\Omega$. Here ${\mathcal A}=\{a_{ij}(x)\}_{ij}$ is a real, Hermitian, uniformly positive definite $N\times N$ matrix; $F\in C^1(\mathbb R^N;\mathbb R^N)$ is Lipschitz continuous; $\beta\in C({\partial}\Omega)$, with $\beta>0$; $\gamma:{\partial}\Omega\times\mathbb R\to \mathbb R; \,q\ge 0$; and ${{{\partial}^{\mathcal A}_{\nu}}} u$ is the conormal derivative of $u$ with respect to $A$; everything is sufficiently regular. Here we prove the well-posedness of the problem. Moreover, we prove explicit stability estimates of the solution $u$ with respect to the coefficients ${\mathcal A},$ $ F,$ $\beta,$ $\gamma,$ $q,$ and the initial condition $f$. Our estimates cover the singular case of a problem with $q=0$ which is approximated by problems with positive $q$.

Studied here are traveling-front solutions $\phi_{\epsilon}(x - {c} t)$ of a conservation law with hyper-dissipation appended. The evolution equation in question is a simple conservation law with a fourth-order dissipative term, namely $$u_t + 2uu_x + {\epsilon} u_{xxxx} = 0 , $$ where ${\epsilon} > 0$. The traveling front is restricted by the asymptotic conditions $\phi_{\epsilon}(x) \to L_\pm$ as $x \to \pm\infty$, where $L_+ < L_-$, and the symmetry condition $\phi_{\epsilon}(x) + \phi_{\epsilon}(-x) = L_- + L_+$ for all $x \in \mathbb R$. Such fronts are shown to exist and proven to be unique. Unlike the corresponding fronts for the Burgers' equation, they do not decay monotonically to their asymptotic states, but oscillate infinitely often around them. Despite this oscillation, it is also shown that $\phi_{\epsilon}(x) \to L_+$ as ${\epsilon} \to 0$, for all $x > 0$, and $\phi_{\epsilon}(x) \to L_-$ as ${\epsilon} \to 0$, for all $x < 0$.

In this article we deal with a class of degenerate parabolic systems which exhibit two phenomena: porous medium effects and chemotaxis ones. Such classes of equations arise in the modelling (on the mesoscale) of biomass spreading mechanisms via chemotaxis. Under certain "balance conditions'' on the order of the porous medium degeneracy and the growth of the chemotactic functions, we prove well posedness.

We present a technique for derivation of a priori bounds for Gevrey--Sobolev norms of space-periodic three-dimensional solutions to evolutionary partial differential equations of hydrodynamic type. It involves a transformation of the flow velocity in the Fourier space, which introduces a feedback between the index of the norm and the norm of the transformed solution, and results in emergence of a mildly dissipative term. We illustrate the technique, using it to derive finite-time bounds for Gevrey--Sobolev norms of solutions to the Euler and inviscid Burgers equations, and global-in-time bounds for the Voigt-type regularizations of the Euler and Navier--Stokes equation (assuming that the respective norm of the initial condition is bounded). The boundedness of the norms implies analyticity of the solutions in space.

In this paper, we study the coupled nonlinear Schrödinger system \begin{equation*} \left\{ \begin{matrix} iu_t+ u_{xx} + (a |u|^2 + b |v|^2) u=0\\ iv_t+ v_{xx} + (b |u|^2 + c |v|^2) v=0\\ \end{matrix} \right. \end{equation*} where $u,v$ are complex-valued functions of $(x,t)\in \mathbb R^2$, and $a,b,c \in \mathbb R$. Our work shows that, for this system of equations, the interplay between components of solutions in terms of the parameters $a,b,c$ plays an important role in both the existence and stability of solitary waves. In particular, we prove that solitary wave solutions to this system are orbitally stable when either $0<b < \min\{a,c\}$, or $b>0$ with $b > \max\{a,c\}$ and $b^2> ac$.