This paper is concerned with the initial boundary value problem for a nonlocal p-Laplacian evolution equation with critical initial energy. In the framework of the energy method, we construct an unstable set and establish its invariance. Finally, the finite time blow-up of solutions is derived by a combination of the unstable set and the concavity method.

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.

We establish the blow-up of solutions to the Kirchhoff equation of q-Laplacian type with a nonlinear dissipative term $\left(\right|u_t{|}^{l-2}{u}_t{)}_t-M\left(\right||A^{1/2}u\left|\right|²₂)Au+|u_t{|}^{\beta }{u}_t={\left|u\right|}^{p}u$, x ∈ Ω, t > 0.

In this paper, we consider the nonlinear Kirchhoff-type equation $$u_{tt}+M\left({∥D^{m}u\left(t\right)∥}_2^2\right){(-\Delta )}^{m}u+{\left|u_t\right|}^{q-2}{u}_t={\left|u_t\right|}^{p-2}u$$ with initial conditions and homogeneous boundary conditions. Under suitable conditions on the initial datum, we prove that the solution blows up in finite time.

We focus on the blow-up in finite time of weak solutions to the wave equation with interior and boundary nonlinear sources and dissipations. Our central interest is the relationship of the sources and damping terms to the behavior of solutions. We prove that under specific conditions relating the sources and the dissipations (namely p > m and k > m), weak solutions blow up in finite time.

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...