An abstract semilinear parabolic equation in a Banach space X is considered. Under general assumptions on nonlinearity this problem is shown to generate a bounded dissipative semigroup on $X^{\alpha }$. This semigroup possesses an $(X^{\alpha }-Z)$-global attractor that is closed, bounded, invariant in $X^{\alpha }$, and attracts bounded subsets of $X^{\alpha }$ in a ’weaker’ topology of an auxiliary Banach space Z. The abstract approach is finally applied to the scalar parabolic equation in Rⁿ and to the partly dissipative system.

This paper is devoted to the study of traveling waves for monotone evolution systems of bistable type. In an abstract setting, we establish the existence of traveling waves for discrete and continuous-time monotone semiflows in homogeneous and periodic habitats. The results are then extended to monotone semiflows with weak compactness. We also apply the theory to four classes of evolution systems.

This paper provides blow up results of Fujita type for a reaction-diffusion system of 3 equations in the form $uₜ-\Delta \left(a_{11}u\right)=h(t,x){\left|v\right|}^p$, $vₜ-\Delta \left(a_{21}u\right)-\Delta \left(a_{22}v\right)=k(t,x){\left|w\right|}^q$, $wₜ-\Delta \left(a_{31}u\right)-\Delta \left(a_{32}v\right)-\Delta \left(a_{33}w\right)=l(t,x){\left|u\right|}^r$, for $x\in {ℝ}^N$, t > 0, p > 0, q > 0, r > 0, $a_{ij}=a_{ij}(t,x,u,v)$, under initial conditions u(0,x) = u₀(x), v(0,x) = v₀(x), w(0,x) = w₀(x) for $x\in {ℝ}^N$, where u₀, v₀, w₀ are nonnegative, continuous and bounded functions. Subject to conditions on dependence on the parameters p, q, r, N and the growth of the functions h, k, l at infinity, we prove finite blow up time for every solution of the above system,...

We prove Fujita-type global existence and nonexistence theorems for a system of m equations (m > 1) with different diffusion coefficients, i.e. $u_{it}-d_i\Delta u_i={\prod }_{k=1}^m{u}_k^{p_k^i},i=1,...,m,x\in {ℝ}^N,t>0,$ with nonnegative, bounded, continuous initial values and $p_k^i\ge 0$, $i,k=1,...,m$, $d_i>0$, $i=1,...,m$. For solutions which blow up at $t=T<\le \infty$, we derive the following bounds on the blow up rate: $u_i(x,t)\le C{(T-t)}^{-{\alpha }_i}$ with C > 0 and ${\alpha }_i$ defined in terms of $p_k^i$.

This paper is mainly concerned with the blow-up and global existence profile for the Cauchy problem of a class of fully nonlinear degenerate parabolic equations with reaction sources.

We present some recent results on the blow-up behavior of solutions of heat equations with nonlocal nonlinearities. These results concern blow-up sets, rates and profiles. We then compare them with the corresponding results in the local case, and we show that the two types of problems exhibit "dual" blow-up behaviors.

This paper deals with the blow-up properties of positive solutions to a localized singular parabolic equation with weighted nonlocal nonlinear boundary conditions. Under certain conditions, criteria of global existence and finite time blow-up are established. Furthermore, when q=1, the global blow-up behavior and the uniform blow-up profile of the blow-up solution are described; we find that the blow-up set is the whole domain [0,a], including the boundary, in contrast to the case of parabolic equations...