A New Critical Behavior for Nonlinear Wave Equations
Tóm tắt
We study the inhomogeneous semilinear wave equations
$$\Delta u{\text{ + }}\left| u \right|^p - u_{tt} + w = 0$$
on
$${\text{M}}^n \times \left( {0,\infty } \right)$$
with initial values
$$u\left( {x,0} \right) = u_0 \left( x \right)$$
and
$$u_t \left( {x,0} \right) = v_0 \left( x \right)$$
,where
$${\text{M}}^n $$
is a noncompact, complete manifold. We founda new critical behavior in the following sense. There exists ap* > 0. When 1 < p ≤ p*, the above problem hasno global solution for any nonnegative
$$w = w\left( x \right)$$
not identicallyzero and for any
$$u_0 $$
and
$$v_0 $$
; when
$$p > p^* $$
the problem has a global solution for some
$$w = w\left( x \right) > 0$$
and some
$$u_0 $$
and
$$v_0 $$
. If
$${\text{M}}^n = {\text{R}}^n $$
, which is equipped with the Euclideanmetric, then
$$p^* = n/\left( {n - 2} \right),n \geqslant 3$$
. If
$$n = 3$$
we show that
$$p^* = 3$$
belongs to the blow upcase. Although homogeneous semilinear wave equations are known to exhibit acritical behavior for a long time, this seems to be the first result oninhomogeneous ones.
Tài liệu tham khảo
P. Baras and M. Pierre, CriteÂre d'existence de solutions positives pour des eÂquations semi-line aires non monotones, Ann. Inst. PoincareÂ, Analyse Non LineÂaire 2, 185–212 (1985).
H. Fujita, On the blowup of solutions of the Cauchy problem for u t=Εu + u, 1.+a; J. Fac. Sci. Univ. Tokyo, Sect. I, 13, 109–124 (1966).
R. Glassey, Finite time blowup for solutions of nonlinear wave equations, Math. Z. 177, 323–340 (1981).
V. Georgiev, H. Lindblad, and C. D. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations, Amer. J. Math. 119, 1291–1319 1997.
F. John, Blowup of solutions of nonlinear wave equations in three dimensions, Man. Math. 28, 235–268 (1979).
T. Kato, Blowup of solutions of some nonlinear hyperbolic equations, Commun. Pure Appl. Math. 32, 501–505 (1980).
H. A. Levine, Instability and nonexistence of global solutions of nonlinear wave equations of the form P utt = – Au + F (u), Trans. AMS. 192, 1–21 (1974).
H. A. Levine, The role of critical exponents in blowup theorems, SIAM Rev. 32, 269–288 (1990).
H. Lindblad and C. D. Sogge, Long-time existence for small amplitude semilinear wave equations, Amer. J. Math. 118, 1047–1135 (1996).
P. Li and S. T. Yau, On the parabolic kernel of the SchroÈ dinger operator, Acta Math. 156, 153–201 (1986).
J. Schaeffer, Finite time blowup for u tt- Εu H (u r; ut). in two space dimensions, Commun. PDE, 11, 513–545 (1986).
T. Sideris, Nonexistence of global solutions of semilinear wave equations in high dimensions, J. Di€. Eq. 52, 378–406 (1984).
W. A. Strauss, Nonlinear wave equations, CBMS AMS 73, (1993).
F. Treves, Basic Linear Partial Di€erential Equations, Academic Press, New York, 1975.
Qi S. Zhang, Nonlinear parabolic problems on manifolds and non-existence result of the non-compact Yamabe problem, Elect. Res. Announce. AMS 3, 45–51 (1997).
Qi S. Zhang, Blowup results for nonlinear parabolic equations on manifolds, Duke Math. J. 97, 515–539.