This paper is concerned with the global existence and asymptotic behaviour, as time tends to infinity, of solutions to the system for a nonlinear viscous, heat-conductive, one-dimensional real gas. Our results show that the global solution approaches to the solution in the $H^1$ norm to the corresponding stationary problem, as time tends to infinity.

We investigate qualitative properties of solutions to the Cauchy problem for the equation $\rho(x)u_t=(u^m)_{xx}-c_0 u^p$, where $m>1$ and $c_0, p >0$; the initial data are nonnegative with compact support and the density $\rho(x)>0$ satisfies suitable decay conditions as $|x|\to\infty$. If $p \ge m$ and $\rho(x)$ decays not faster than $|x|^{-k}$, where $0 <k \le k^*:=2(p-1)/(p-m)$, the interfaces exist globally in time. On the contrary, if $\rho(x)$ decays faster than $|x|^{-k}$ with $k>k^*$, the interfaces can disappear in finite time. It is also proved that solutions go to zero uniformly as $t \to\infty $, at variance from the case $c_0=0$.

We study a one dimensional Dirichlet problem of fourth order and a corresponding "buckling eigenvalue problem" under Dirichlet boundary conditions. These problems may serve as model problems for "Orr-Sommerfeld" like boundary and eigenvalue problems. It turns out that eigenvalue curves in appropriate parameter domains look completely different than for the same equation under so called Navier boundary conditions. Further emphasis is laid on positivity properties, and also here, fundamental differences with Navier conditions arise: It may e.g. happen that one has infinitely many linearly independent positive eigenfunctions. Connections and analogies with the clamped plate boundary value problem on families of deformed domains are discussed.

In this paper we prove that the equation ${du \over dt} + \sum_{i=0}^n a_i(t)u^i = f(t),$ $t \in [0,1]$, $u(0) = u(1)$, has for every continuous $f$ at most $n$ solutions provided that $n$ is odd, and the continuous coefficients $a_i$ satisfy $|a_n(t)| \ge \alpha > 0$ and $|a_i(t)| \le \beta,$ $i = 1,\dots,n-1$, with $\beta > 0$ sufficiently small. Furthermore, we show that this result implies that for a restricted subclass of polynomial vector fields of order $n$ in $\mathbb R^2$ the maximal number of limit cycles is $n$. This constitutes a special case of Hilbert's 16th problem.

In this paper we study a class of nonlinear integro-differential equations which correspond to a fractional-order time derivative and interpolate nonlinear heat and wave equations. For this purpose we first establish some space--time estimates of the linear flow which is produced by Mittag--Leffler's functions based on Mihlin--Hörmander's multiplier estimates and other harmonic analysis tools. Using these space--time estimates we prove the well-posedness of a local mild solution of the Cauchy problem for the nonlinear integro-differential equation in $ C([0,T); L^p(\mathbf R^n))$ or $L^q(0, T; L^p(\mathbf R^n))$.

The purpose of this paper is to prove an existence and uniqueness theorem of positive solutions of diffusive logistic equations with indefinite weights, which model population dynamics in environments with strong spatial heterogeneity. We prove that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment. Moreover we discuss the stability properties for positive steady states.