On potential wells and applications to semilinear hyperbolic equations and parabolic equations

Global existence, blow up and asymptotic behaviour of solutions for nonlinear Klein–Gordon equation with dissipative term

“…By the argument in [27] we can obtain the following Theorems 3.1-3.3 and Lemma 3.4. Theorem 3.1 shows the invariance of W and V , respectively.…”

Blow-up and non-extinction for a nonlocal parabolic equation with logarithmic nonlinearity

“…Because of this main difficulty, some most effective methods, such as the method of upper and lower solutions, are not valid here anymore. Inspired by the ideas in [9][10][11][12][13][14][15][16][17][18], the threshold results for the global existence and blow-up of the weak solutions are given by the potential well method, the classical Galerkin method, and the logarithmic Sobolev inequality. Further we discuss the non-extinction properties and the asymptotic behavior of the global solutions.…”

Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data

“…Of course that is correct, but on page 294 of [11], near the top of the page, Payne and Sattinger rule out this possibility. The remark in [21] may leave the impression that they did not consider this possibility.…”

“…Recently, some other incorrect proofs regarding both invariance of the set V and global non-existence (see Lemma 4.2, Theorem 4.3 and Theorem 6.3 in [11]) were pointed out and corrected in [21] by introducing a family of potential wells. In the correction (ii) on page 2667 of [21], the authors indicate that M (t) could always be negative. Of course that is correct, but on page 294 of [11], near the top of the page, Payne and Sattinger rule out this possibility.…”

“…Moreover, in [21], some new results were achieved on the invariant sets, specifically global existence of solution for the critical initial data [21] are the following theorems:…”