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 investigate critical exponents for blow-up of nonnegative solutions to a class of parabolic inequalities. The proofs make use of a priori estimates of solutions combined with a simple scaling argument.

This paper deals with the blow-up properties of the non-Newtonian polytropic filtration equation $u_t-div\left(\right|\nabla u^m{|}^{p-2}\nabla u^m)=f\left(u\right)$ with homogeneous Dirichlet boundary conditions. The blow-up conditions, upper and lower bounds of the blow-up time, and the blow-up rate are established by using the energy method and differential inequality techniques.

Consider the nonlinear heat equation (E): $u_t-\Delta u={\left|u\right|}^{p-1}u+b{\left|\nabla u\right|}^q$. We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates $C₁{(T-t)}^{-1/(p-1)}{\le \left|\right|u\left(t\right)\left|\right|}_{\infty }\le C₂{(T-t)}^{-1/(p-1)}$. Also, as an application of our method, we obtain the same upper estimate if u only satisfies the nonlinear parabolic inequality $u_t-u_{xx}\ge u^p$. More general inequalities of the form $u_t-u_{xx}\ge f\left(u\right)$ with, for instance, $f\left(u\right)=(1+u)lo{g}^p(1+u)$ are also treated. Our results show that for solutions of the parabolic inequality, one has essentially the same estimates as for solutions of the ordinary...

In this paper, we will consider blowup solutions to the so called Keller-Segel system and its simplified form. The Keller-Segel system was introduced to describe how cellular slime molds aggregate, owing to the motion of the cells toward a higher concentration of a chemical substance produced by themselves. We will describe a common conjecture in connection with blowup solutions to the Keller-Segel system, and some results for solutions to simplified versions of the Keller-Segel system, giving the...

We give necessary and sufficient conditions for the formal power series solutions to the initial value problem for the Burgers equation ${\partial }_{t}u-{\partial }_x²u={\partial }_x\left(u²\right)$ to be convergent or Borel summable.