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 discuss the effect of time delay on blow-up of solutions to initial-boundary value problems for nonlinear reaction-diffusion equations. Firstly, two examples are given, which indicate that the delay can both induce and prevent the blow-up of solutions. Then we show that adding a new term with delay may not change the blow-up character of solutions.

In this paper, we show finite time blow-up of solutions of the p−wave equation in ℝN, with critical Sobolev exponent. Our work extends a result by Galaktionov and Pohozaev [4]

A class of nonlinear viscous transport equations describing aggregation phenomena in biology is considered. General conditions on an interaction potential are obtained which lead either to the existence or to the nonexistence of global-in-time solutions.

We study necessary and sufficient conditions for the existence of nonnegative boundary blow-up solutions to the cooperative system ${\Delta }_{p}u=g(u-\alpha v),{\Delta }_{p}v=f(v-\beta u)$ in a smooth bounded domain of ${ℝ}^N$, where ${\Delta }_p$ is the p-Laplacian operator defined by ${\Delta }_{p}u={div\left(\right|\nabla u|}^{p-2}\nabla u)$ with p > 1, f and g are nondecreasing, nonnegative C¹ functions, and α and β are two positive parameters. The asymptotic behavior of solutions near the boundary is obtained and we get a uniqueness result for p = 2.

Let $\Omega$ be a bounded domain in ${ℝ}^n$ with smooth boundary $\partial \Omega$. We consider the equation $d^2\Delta u-u+u^{\frac{n-k+2}{n-k-2}}=0\text{in}\Omega$, under zero Neumann boundary conditions, where $\Omega$ is open, smooth and bounded and $d$ is a small positive parameter. We assume that there is a $k$-dimensional closed, embedded minimal submanifold $K$ of $\partial \Omega$, which is non-degenerate, and certain weighted average of sectional curvatures of $\partial \Omega$ is positive along $K$. Then we prove the existence of a sequence $d=d_j\to 0$ and a positive solution $u_d$ such that $d^2{\left|\nabla u_d\right|}^2\rightharpoonup S{\delta }_K\mathrm{as}d\to 0$ in the sense of measures, where ${\delta }_K$...

This paper is devoted to the study of cloaking via anomalous localized resonance (CALR) in the two- and three-dimensional quasistatic regimes. CALR associated with negative index materials was discovered by Milton and Nicorovici [21] for constant plasmonic structures in the two-dimensional quasistatic regime. Two key features of this phenomenon are the localized resonance, i.e., the fields blow up in some regions and remain bounded in some others, and the connection between the localized resonance...

On the unit disk $B_1\subset {ℝ}^2$ we study the Moser-Trudinger functional $E\left(u\right)={\int }_{B_1}\left(e^{u^2}-1\right)dx,u\in H_0^1\left(B_1\right)$ and its restrictions ${E|}_{M_{\Lambda }}$, where $M_{\Lambda }:=\{u\in H_0^1\left(B_1\right){:\parallel u\parallel }_{H_0^1}^2=\Lambda \}$ for $\Lambda >0$. We prove that if a sequence $u_k$ of positive critical points of ${E|}_{M_{{\Lambda }_k}}$ (for some ${\Lambda }_k>0$) blows up as $k\to \infty$, then ${\Lambda }_k\to 4\pi$, and $u_k\to 0$ weakly in $H_0^1\left(B_1\right)$ and strongly in $C_{\mathrm{loc}}^1({\overline{B}}_1\setminus \left\{0\right\})$. Using this fact we also prove that when $\Lambda$ is large enough, then ${E|}_{M_{\Lambda }}$ has no positive critical point, complementing previous existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe.

We consider the Cauchy problem for the focusing Hartree equation $i{u}_t+\Delta u+{\left(\right|·|}^{-3}\ast \left|u\right|²)u=0$ in ℝ⁵ with initial data in H¹, and study the divergence property of infinite-variance and nonradial solutions. For the ground state solution of $-Q+\Delta Q+\left(\right|·{|}^{-3}\ast \left|Q\right|²)Q=0$ in ℝ⁵, we prove that if u₀ ∈ H¹ satisfies M(u₀)E(u₀) < M(Q)E(Q) and ||∇u₀||₂||u₀||₂ > ||∇Q||₂||Q||₂, then the corresponding solution u(t) either blows up in finite forward time, or exists globally for positive time and there exists a time sequence tₙ → ∞ such that ||∇u(tₙ)||₂ → ∞....

We consider the existence, both locally and globally in time, the decay and the blow up of the solution for the extensible beam equation with nonlinear damping and source terms. We prove the existence of the solution by Banach contraction mapping principle. The decay estimates of the solution are proved by using Nakao’s inequality. Moreover, under suitable conditions on the initial datum, we prove that the solution blow up in finite time.

This paper deals with existence of finite-time blow-up solutions to a degenerate parabolic–elliptic Keller–Segel system with logistic source. Recently, finite-time blow-up was established for a degenerate Jäger–Luckhaus system with logistic source. However, blow-up solutions of the aforementioned system have not been obtained. The purpose of this paper is to construct blow-up solutions of a degenerate Keller–Segel system with logistic source.