On the existence or the absence of global solutions of the Cauchy characteristic problem for some nonlinear hyperbolic e
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For wave equations with power nonlinearity we investigate the problem of the existence or nonexistence of global solutions of the Cauchy characteristic problem in the light cone of the future. 1. Statement of the problem Consider a nonlinear wave equation of the type u :=
∂2 u − ∆u = f (u) + F, ∂t 2
(1.1)
where f and F are the given real functions; note that f is a nonlinear and u is an unknown real function, ∆ = ni=1 ∂2 /∂xi2 . For (1.1), we consider the Cauchy characteristic problem on finding in a truncated light cone of the future DT : |x| < t < T, x = (x1 ,...,xn ), n > 1, T = const > 0, a solution u(x,t) of that equation by the boundary condition u|ST = g,
(1.2)
where g is the given real function on the characteristic conic surface ST : t = |x|, t ≤ T. When considering the case T = +∞ we assume that D∞ : t > |x| and S∞ = ∂D∞ : t = |x|. Note that the questions on the existence or nonexistence of a global solution of the Cauchy problem for semilinear equations of type (1.1) with initial conditions u|t=0 = u0 , ∂u/∂t |t=0 = u1 have been considered in [1, 2, 6, 7, 8, 10, 13, 14, 15, 16, 17, 18, 22, 23, 26, 30, 31]. As for the characteristic problem in a linear case, that is, for problem (1.1)-(1.2) when the right-hand side of (1.1) does not involve the nonlinear summand f (u), this problem is, as is known, formulated correctly, and the global solvability in the corresponding spaces of functions takes place [3, 4, 5, 11, 25]. Below we will distinguish the particular cases of the nonlinear function f = f (u), when problem (1.1)-(1.2) is globally solvable in one case and unsolvable in the other one. Copyright © 2006 Hindawi Publishing Corporation Boundary Value Problems 2005:3 (2005) 359–376 DOI: 10.1155/BVP.2005.359
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The Cauchy characteristic problem
2. Global solvability of the problem Consider the case for f (u) = −λ|u| p u, where λ = 0 and p > 0 are the given real numbers. In this case (1.1) takes the form Lu :=
∂2 u − ∆u = −λ|u| p u + F, ∂t 2
(2.1)
where for convenience we introduce the notation L = . As is known, (2.1) appears in the relativistic quantum mechanics [13, 24, 28, 29]. For the sake of simplicity of our exposition we will assume that the boundary condition (1.2) is homogeneous, that is, u|ST = 0.
(2.2)
◦
Let W21 (DT ,ST ) = {u ∈ W21 (DT ) : u|ST = 0}, where W21 (DT ) is the known Sobolev space. ◦
Remark 2.1. The embedding operator I : W21 (DT ,ST ) → Lq (DT ) is the linear continuous compact operator for 1 < q < 2(n + 1)/(n − 1) when n > 1 [21, page 81]. At the same time, Nemytski’s operator K : Lq (DT ) → L2 (DT ), acting by the formula Ku = −λ|u| p u, is continuous and bounded if q ≥ 2(p + 1) [19, page 349], [20, pages 66–67]. Thus if p < 2/(n − 1), that is, 2(p + 1) < 2(n + 1)/(n − 1), then there exists the number q such that 1 < 2(p + 1) ≤ q < 2(n + 1)/(n − 1), and hence the operator ◦
K0 = KI : W21 DT ,ST −→ Lq DT
(2.3)
◦
is continuous and compact and, more so, from u ∈ W21 (DT ,ST ) follows u ∈ L p+1 (DT ). As is mentioned above, here, and in the sequel
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