We study the asymptotic blow-up behavior of nonnegative solutions to the quasilinear heat equation $$ u_t = (u^2)_{xx} + u^2 \quad \text{for $x \in \mathbf{R}, \,\, t > 0$}, $$ with nonnegative, bounded, continuous initial data. We give a complete classification of all possible types of blow-up behavior for compactly supported initial data. For data which look like a step function we construct self-similar blow-up patterns (logarithmic traveling wave solutions) for which the corresponding blow-up sets are empty.

We study the asymptotic blow-up behavior of nonnegative solutions to the quasilinear heat equation $$ u_t = (u^2)_{xx} + u^2 \quad \text{for $x \in \mathbf{R}, \,\, t > 0$}, $$ with nonnegative, bounded, continuous initial data. We give a complete classification of all possible types of blow-up behavior for compactly supported initial data. For data which look like a step function we construct self-similar blow-up patterns (logarithmic traveling wave solutions) for which the corresponding blow-up sets are empty.

Periodic parabolic reaction-diffusion equations on $\mathbb{R}^N$ are considered. Using the method of sub- and supersolutions time-periodic solutions are obtained, which are stable with respect to the $L_\infty$-norm. Particular attention is devoted to Fisher's equation from population genetics.

This paper deals with a parabolic partial differential equation that incorporates diffusion and convection terms and that previously has been shown to have solutions that become unbounded at a single point in finite time. The results presented here describe the limiting behavior of the solution in a neighborhood of the blowup point, as well as the asymptotic growth rate as the blowup time is approached. Rigorous estimates are proved, and some supplementary numerical calculations are presented.

In [2] we proved global boundedness for the system (1.1); see Theorem 1.1 below. Here we show global existence for weak solutions (Theorem 1.2), existence of a global bounded attractor (Theorem 1.3), and strong continuous dependence on the data (Theorem 1.4) for the weak solutions.

The solvability of the periodic problem $$ v'(t)+\mathcal{B}(t)u(t)+\mathcal {C}(t)u(t)\ni \tilde f(t), \ \ v(t)\in A(t)u(t),\ \, v(0)=v(T), $$ will be investigated. It will be shown that the solution exists if $\mathcal {B}t)$ are maximal monotone operators, $\mathcal {A}(t)$ are subdifferentials, compact and possibly degenerated, $\mathcal {C}(t)$ are Lipschitzian and compact, and $\mathcal{B}(t)+\mathcal {C}(t)$ are coercive. The proof consists of several limit processes for an approximating periodic problem which has a solution.

We are concerned with the asymptotic behavior of global solutions for a class of reaction-diffusion systems under homogeneous Neumann boundary conditions. An example of the system which we consider in this paper is what we call a diffusive epidemic model. After we show that every global solution uniformly converges to the corresponding constant function as $t \to \infty$, we investigate the rate of this convergence. We can obtain it with use of $L^p$-estimates, integral equations via analytic semigroups, fractional powers of operators and some imbedding relations.

In this paper we study ordinary differential equations on a noncomplete Riemannian manifold. Using the Ljusternik-Schnirelmann theory, we prove the existence of infinitely many solutions joining two given submanifolds.

We prove the existence and uniqueness of global smooth solutions to the Dirichlet initial-boundary problem in nonlinear $n$-dimensional viscoelasticity of integral type for small initial data in $H^{s_0}(\Omega)$, where $s_0$ is an integer smaller than that needed to establish the local existence. Moreover, the exponential decay of the solution is obtained.

We study well-posedness in $L^P$ of the Cauchy problem for second order parabolic equations with time-independent measurable coefficients by means of constructing corresponding Cosernigroups. Lower order terms are considered as form-bounded perturbations of the generator of the symmetric submarkovian sernigroup associated with the Dirichlet form. It is shown that the Cosernigroup corresponding to the Cauchy problem exists in a certain interval in the scale of $L^P$-spaces which depends only on form-bounds of perturbations. We establish also analyticity and $L^P$ -smoothness of the sernigroup constructed.

We study a convective problem of "mixed type" (i.e., parabolic and hyperbolic ) which has a degenerate second order term. The analysis by the "vanishing viscosity" is made by a careful study of the phase boundaries and of the topological structure of the phase sets. At the end we provide the existence of a discontinuous travelling wave.

In this paper we show that for a convenient choice of the nonlinear map $a(u)$ the equation $u_t + \div (a (u)) - \Delta u_t = 0$ has traveling waves solution $\phi_c (x - \overrightarrow{c} t)$, where $\overrightarrow{c} = (c,\ldots, c)\in\mathbb{R}^n$. For $c$ varying in a suitable interval we show that these traveling waves are stable.