Abstract
In this paper we establish the existence of certain classes of solutions to the energy critical nonlinear wave equation in dimensions 3 and 5 assuming that the energy exceeds the ground state energy only by a small amount. No radial assumption is made. We find that there exist four sets in H(over dot) x L-2 with nonempty interiors which correspond to all possible combinations of finite-time blowup on the one hand, and global existence and scattering to a free wave, on the other hand, as t -> +/-infinity.
Highlights
Consider the energy-critical nonlinear wave equation with real-valued u u − ∆u = |u|p−1u, u(t, x) : R1+d → R, p = d + 2 = 2∗ − 1, d−2 d = 3 or 5, (1.1)in the energy space u(t) := (u(t), u (t)) ∈ H := H 1(Rd) × L2(Rd), (1.2)which is the real Hilbert space with the inner product u, v H := ∇u1|∇v1 + u2|v2, f |g := f (x) · g(x) dx
U = (u, u ) denotes the vector derived from a time function u(t), while a general vector is denoted like u = (u1, u2)
In this paper we study the behavior of solutions with
Summary
Note that the derivative of J(φ) with respect to any scaling φ(x) → eaσφ(ebσx) except for S−σ 1 gives a non-zero constant multiple of K(φ) This is a special feature of the scaling critical case, which allows us to work with a single K, whereas in the subcritical case [21] we needed two different functionals and their equivalence. The solution W+ is the one discovered by Duyckaerts, Merle [7] It is a radial H 1 × L2 solution, exists globally in forward time and approaches W in H 1, and blows up in finite negative time. As noted in [5, Remark 6.5], the removal of the radial assumption in [12] would complete [7] in the sense that the L2-condition can be removed even nonradially This is what we accomplish in this paper, whence Corollary 1.3. We refer the reader to [5]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
