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This paper deals with the following mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity u t − Δ u t − d i v (| ∇ u | p − 2 ∇ u) = | u | q − 2 u log ⁡ | u | in a bounded domain with zero Dirichlet boundary condition, which was studied in our previous paper [J Math Anal Appl. 2019;478(2):393-420]. In view the results of [J Math Anal Appl. 2019;478(2):393-420], for the case (1) 1 < p ≤ q ≤ 2 , i f n ≤ p , ≤ 2 , i f 2 n n + 2 < p < n , < n p n − p , i f p ≤ 2 n n + 2 , the global existence and blow-up results were got when J (u 0) ≤ d , where d denotes the mountain-pass level. But for the case (2) 1 < p ≤ q a n d 2 < q < ∞ , i f n ≤ p , n p n − p , i f 2 n n + 2 < p < n , the blow-up results were got when J (u 0) ≤ M , where M ≤ d is a constant. In this paper, we extend and complete the results of [J Math Anal Appl. 2019;478(2):393-420] on the following three aspects: First, the blow-up results are got when J (u 0) ≤ d and (2) are satisfied. Second, the upper and lower bounds of blow-up time are estimated. Third, the global existence and blow-up results are got when J (u 0) > d. [ABSTRACT FROM AUTHOR]

This paper presents asymptotic formulas in the case of the following two problems for the Pucci's extremal operators M ± . It is considered the solution u ε (x) of − ε 2 M ± ∇ 2 u ε + u ε = 0 in Ω such that u ε = 1 on Γ. Here, Ω ⊂ R N is a domain (not necessarily bounded) and Γ is its boundary. It is also considered v (x , t) the solution of v t − M ± ∇ 2 v = 0 in Ω × (0 , ∞) , v = 1 on Γ × (0 , ∞) and v = 0 on Ω × { 0 }. In the spirit of their previous works [Berti D, Magnanini R. Asymptotics for the resolvent equation associated to the game-theoretic p-laplacian. Appl Anal. 2019;98(10):1827–1842.; Berti D, Magnanini R. Short-time behavior for game-theoretic p-caloric functions. J Math Pures Appl (9). 2019;(126):249–272.], the authors establish the profiles as ϵ or t → 0 + of the values of u ε (x) and v (x , t) as well as of those of their q-means on balls touching Γ. The results represent a further step in the extensions of those obtained by Varadhan and by Magnanini-Sakaguchi in the linear regime. [ABSTRACT FROM AUTHOR]

In this paper, we study the evolution p, q-Laplacian equations u t = d i v (| ∇ u | p − 2 ∇ u) + u α ∫ Ω v m d x and v t = d i v (| ∇ v | q − 2 ∇ v) + v β ∫ Ω u n d x with 1 (p − 1 − α) (q − 1 − β) , there exist suitable initial data such that vanishing solutions exist. If m n < (p − 1 − α) (q − 1 − β) , we find the explicit scopes of initial data such that the solutions could not vanish, which complete the corresponding classifications of solutions in Math. Methods Appl. Sci. 39 (2016) 1325–1335 and Appl. Math. Comp. 259 (2015) 587–595, respectively. For the critical case m n = (p − 1 − α) (q − 1 − β) , the solutions vanish in finite time with small initial data. [ABSTRACT FROM AUTHOR]