Search

In this paper, we consider the prescribed Q -curvature equation with a singularity Δ 2 u = | x | − α K (x) e 4 u in R 4 , Λ = ∫ R 4 | x | − α K (x) e 4 u d x < + ∞. First, we prove that all solutions are radially symmetric for 0 < α < 4 , K > 0 constant and Λ = 16 π 2 (1 − α 4). Next, some blow-up results have been studied by using a classification result for K > 0 constant and 0 < α < 4. Finally, we show that equation with K (x) = (1 − | x | p) and 0 < α < 4 has normal solutions (namely solutions which can be written in integral form) if and only if p ∈ (0 , 4 − α) and 8 π 2 (1 + p − α 4) ≤ Λ < 16 π 2 (1 − α 4). [ABSTRACT FROM AUTHOR]

In this paper we mainly investigate the initial value problem of the periodic Euler-Poincaré equations. We first present a new blow-up result to the system for a special class of smooth initial data by using the rotational invariant properties of the system. Then, we prove that the periodic Euler-Poincaré equations are ill-posed in critical Besov spaces by a contradiction argument. Finally, we verify the system possesses a class of peakon solutions in the sense of distributions. [ABSTRACT FROM AUTHOR]

In this paper, we study the blow-up mechanism and persistence property of solutions to the modified b-family of equations. The dynamics of the blow-up quantity along the characteristics is established by the Riccati-type differential inequality with various parameters. The key feature of the method is to refine the analysis on the growth rate of the relative ratio between solution and its gradient by performing a vertical shift. Furthermore, the persistence results for the solution are established in weighted spaces. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the authors study the equation in with . We first prove that for , the solution exists at least for a short time; then for , the existence and nonexistence of global (in time) solutions are studied in various situations. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the positive solution of the Cauchy problem for the equationand give a secondary critical exponent of the behavior of initial value at infinity for the existence of global and nonglobal solutions of the Cauchy problem. Furthermore, the life span of solutions are also studied. [Copyright &y& Elsevier]