Blow-up in Nonlinear Sobolev Type Equations

On wave equation: review and recent results

“…Also, they obtained an explicit and general decay rate, depending on σ, g and φ, for the energy of solutions of (1.1) without any growth assumption on g and φ at the origin, and on σ at infinity. Also, the following problem 2) where is a bounded region in R n (n ≥ 1), with a smooth boundary ∂ , was considered by many authors. For instance, in the case when f (u) = |u| p−2 u, g(u t ) = |u t | m−2 u t , m, p > 2, Nakao [70] showed that (1.2) has a unique global weak solution if 0 ≤ p − 2 ≤ 2/(n − 2), n ≥ 3 and a global unique strong solution if p −2 > 2/(n −2), n ≥ 3.…”

“…Vitillaro [87] combined the arguments in [45] and [26] to extend these results to situations where the damping is nonlinear and the solution has positive initial energy. For more results concerning blowup and nonexistence, we mention here the work of Vitillaro [88], Todorova [84], Todorova and Vitillaro [85], Wang [89], Liu [51], Wu [90], and the very recent book of Al'shin et al [2]. For the nonlinear Kirchhoff-type problem of the form…”

“…We consider the initial-boundary value problem (P) on the domain B(0, 1) × (0, T ), where B(0, 1) is the unit disk in R 2 . We take the initial data u 0 (x 1 , x 2 ) = 1 − x 2 1 − x 2 2 and u 1 (x 1 , x 2 ) = 0. We consider the following applications: 1.…”

Blowup of a positive-energy solution of model wave equations in nonlinear dynamics

“…В упомянутых работах используется классический метод Левина. В настоящей работе мы применяем модифицированный метод Левина, раз-витый в работе [5], который позволит нам получить всего два основных условия на начальные функции; кроме того, мы легко проверим совместность всех полученных условий.…”

“…Используя технику работы [5], можно перейти к пределу при m → +∞ в нера-венстве (29) и получить неравенство Φ(t) Φ −1/4 (0) − At −1/4 , где…”