Local existence and global nonexistence theorems for a damped nonlinear hyperbolic equation

Global Nonexistence in a Nonlinearly Damped Wave Equation

Existence and Nonexistence of Global Solutions of a Fully Nonlinear Parabolic Equation

In the paper, we study the global existence of weak solution of the fully nonlinear parabolic problem (1.1)-(1.3) with nonlinear boundary conditions for the situation without strong absorption terms. Also, we consider the blow up of global solution of the problem (1.1)-(1.3) by using the convexity method.

Existence, global nonexistence, and asymptotic behavior of solutions for the Cauchy problem of a multidimensional generalized damped Boussinesq-type equation

where u (x, t) denotes the unknown function, f (s) is the given nonlinear function, u0 (x) and u1 (x) are the given initial value functions, k is a constant, the subscript t indicates the partial derivative with respect to t, n is the dimension of space variable x, and △ denotes the Laplace operator in R. The effects of small nonlinearity and dispersion are taken into consideration in the deriv...

Nonexistence of Global Solutions of a Nonlinear Hyperbolic System

Consider the initial value problem utt = ∆u+ |v|p, vtt = ∆v + |u|q, x ∈ Rn, t > 0, u(x, 0) = f(x), v(x, 0) = h(x), ut(x, 0) = g(x), vt(x, 0) = k(x), with 1 ≤ n ≤ 3 and p, q > 0. We show that there exists a bound B(n) (≤ ∞) such that if 1 < pq < B(n) all nontrivial solutions with compact support blow up in finite time.

Global Existence and Asymptotic Behavior of Solutions for Some Nonlinear Hyperbolic Equation

The initial boundary value problem for a class of hyperbolic equation with nonlinear dissipative term u tt − n i1 ∂/∂x i |∂u/∂x i | p−2 ∂u/∂x i a|u t | q−2 u t b|u| r−2 u in a bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set in W 1,p 0 ΩΩ and show the asymptotic behavior of the global solutions through the use of an important l...