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Some Additional Remarks on the Nonexistence of Global Solutions to Nonlinear Wave Equations
Let P and A be symmetric linear operators defined on a dense domain $D \subset H$, a (real) Hilbert space. Let $(x,Av) \geqq \lambda (x,x)$ for all $x \in D$ and some $\lambda > 0$ and $(x,Px) > 0$
A System of Reaction Diffusion Equations Arising in the Theory of Reinforced Random Walks
It is shown that under some circumstances, finite-time blow-up of solutions is possible and in other circumstances, the solutions will decay to a spatially constant solution (collapse).
The Role of Critical Exponents in Blowup Theorems
In this article various extensions of an old result of Fujita are considered for the initial value problem for the reaction-diffusion equation $u_t = \Delta u + u^p $ in $R^N $ with $p > 1$ and
The Role of Critical Exponents in Blow-Up Theorems: The Sequel
Abstract In [ 27 ] Fujita showed that for positive solutions, the initial value problem (in R N ) for u t  = Δ u  +  u p with p  > 1 exhibited the following behavior: If p p c  ≡ 1 + 2/ N , then the
On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains
In this paper we study the first initial-boundary value problem for ut = Au + uP in conical domains D = (0, oo) x Q c RN where Q c SN-I is an open connected manifold with boundary. We obtain some
Unrestricted lower bounds for eigenvalues for classes of elliptic equations and systems of equations with applications to problems in elasticity
Lower bounds for the eigenvalues of some elliptic equations and elliptic systems over bounded regions are obtained. The bounds are universal in that they depend only upon the volume of the region.
Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy
In this paper we consider the long time behavior of solutions of the initial value problem for semi-linear wave equations of the form utt + alutlmlut Au = bluIP-lu in [0, oo) x R'. Here a, b > 0. We