scholarly journals LYAPUNOV FUNCTIONAL AND BLOW-UP RESULTS FOR A CLASS OF PERTURBATIONS OF SEMILINEAR WAVE EQUATIONS IN THE CRITICAL CASE

2012 ◽  
Vol 09 (02) ◽  
pp. 195-221 ◽  
Author(s):  
MOHAMED ALI HAMZA ◽  
HATEM ZAAG
Keyword(s):  
Wave Equation ◽  
Singular Solution ◽  
Lyapunov Functional ◽  
Wave Equations ◽  
Critical Case ◽  
Critical Power ◽  
Blow Up ◽  
Semilinear Wave Equation ◽  
Nirenberg Inequality ◽  
Blow Up Rate

We study a class of perturbations for the semilinear wave equation with critical power nonlinearity (in the conformal transform sense). Working in the framework of similarity variables, we introduce a Lyapunov functional for this problem. Using a two-step argument based on interpolation and a critical Gagliardo–Nirenberg inequality, we establish that the blow-up rate of any singular solution is given by the solution of the nonperturbed associated ODE, specifically u″ = up.

Global well-posedness and self-similarity for semilinear wave equations in a time-weighted framework of Besov type

2020 ◽  
Vol 17 (01) ◽  
pp. 123-139
Author(s):  
Lucas C. F. Ferreira ◽  
Jhean E. Pérez-López
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Besov Spaces ◽  
Wave Equations ◽  
Well Posedness ◽  
Self Similarity ◽  
Semilinear Wave Equations ◽  
Semilinear Wave Equation

We show global-in-time well-posedness and self-similarity for the semilinear wave equation with nonlinearity [Formula: see text] in a time-weighted framework based on the larger family of homogeneous Besov spaces [Formula: see text] for [Formula: see text]. As a consequence, in some cases of the power [Formula: see text], we cover a initial-data class larger than in some previous results. Our approach relies on dispersive-type estimates and a suitable [Formula: see text]-product estimate in Besov spaces.

Extremal functions for the Moser–Trudinger inequality of Adimurthi–Druet type in W1,N(ℝN)

2019 ◽  
Vol 21 (04) ◽  
pp. 1850023 ◽  
Author(s):  
Van Hoang Nguyen
Keyword(s):  
Critical Case ◽  
Blow Up ◽  
Sharp Inequality ◽  
Extremal Functions ◽  
Analysis Method ◽  
Subcritical Case ◽  
Best Constant ◽  
Blow Up Analysis ◽  
Nirenberg Inequality ◽  
Trudinger Inequality

We study the existence and nonexistence of maximizers for variational problem concerning the Moser–Trudinger inequality of Adimurthi–Druet type in [Formula: see text] [Formula: see text] where [Formula: see text], [Formula: see text] both in the subcritical case [Formula: see text] and critical case [Formula: see text] with [Formula: see text] and [Formula: see text] denotes the surface area of the unit sphere in [Formula: see text]. We will show that MT[Formula: see text] is attained in the subcritical case if [Formula: see text] or [Formula: see text] and [Formula: see text] with [Formula: see text] being the best constant in a Gagliardo–Nirenberg inequality in [Formula: see text]. We also show that MT[Formula: see text] is not attained for [Formula: see text] small which is different from the context of bounded domains. In the critical case, we prove that MT[Formula: see text] is attained for [Formula: see text] small enough. To prove our results, we first establish a lower bound for MT[Formula: see text] which excludes the concentrating or vanishing behaviors of their maximizer sequences. This implies the attainability of MT[Formula: see text] in the subcritical case. The proof in the critical case is based on the blow-up analysis method. Finally, by using the Moser sequence together with the scaling argument, we show that MT[Formula: see text]. Our results settle the questions left open in [J. M. do Ó and M. de Souza, A sharp inequality of Trudinger–Moser type and extremal functions in [Formula: see text], J. Differential Equations 258 (2015) 4062–4101; Trudinger–Moser inequality on the whole plane and extremal functions, Commun. Contemp. Math. 18 (2016) 32 pp.].

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