Consider the following Cauchy problem for the first order quasilinear strictly hy- perbolic system ?u ?u + A(u) = 0, ?t ?x t = 0 : u = f(x). We let M = sup |f (x)| < +∞. x∈R The main result of this paper is that under the assumption that the system is weakly linearly degenerated, there exists a positive constant ε independent of M, such that the above Cauchy problem admits a unique global C1 solution u = u(t,x) for all t ∈ R, provided that +∞ |f (x)|dx ≤ ε, ?∞ +∞ ε |f(x)|dx ≤ . M∞
In this paper, we prove local and global existence of classical solutions for a system of equations concerning an incompressible viscoelastic fluid of Oldroyd-B type via the incompressible limit when the initial data are sufficiently small.
Zhen LEI School of Mathematical Sciences, Fudan University, Shanghai 200433, China
In this paper, the author considers equations with critical exponent in n ≥4 space tions on the initial data, it is proved that there small the initial data are. the Cauchy problem for semilinear wave dimensions. Under some positivity condican be no global solutions no matter how
In this paper, we study how much regularity of initial data is needed to ensure existence of a local solution to the following semilinear wave equations utt-△u=F（u,Du） u（0,x）=f（x）∈H^s,δtu（0,x）=g（x）∈H^s-1,where F is quadratic in Du with D = （δr, δx1,…, δxn）.We proved that the range of s is s ≥n＋1/2 ＋ δ, respectively, with δ 〉 1/4 if n = 2, and δ 〉 0 if n = 3, and δ ≥0 if n ≥ 4. Which is consistent with Lindblad＇s counterexamples [3] for n = 3, and the main ingredient is the use of the Strichartz estimates and the refinement of these.
