ABSTRACT
In the previous chapters we have introduced some useful methods to deal with the issue of global existence and uniqueness of the solution to nonlinear evolution equations with arbitrarily given initial data. However, for a given nonlinear evolution equation, there does not always exist a global solution for arbitrarily given initial data. To see this, in addition to the examples in Chapter 1, let us consider the following initial boundary value problem for a simple nonlinear parabolic equation:
∂u
∂t −∆u = u3 − u, (x, t) ∈ Ω×R+,
u|Γ = 0,
u(0) = φ(x)
where Ω ⊂ Rn is a bounded domain with a smooth boundary Γ. In what follows we show that when initial data φ are sufficiently large, then the solution to problem (5.1.1) must blow up in a finite time. Indeed, multiplying the equation in (5.1.1) by u, then integrating over Ω, we get
1 2 d
dt ‖u(t)‖2 +
∫ Ω (|∇u|2 − |u|4 + |u|2)dx = 0. (5.1.2)
Throughout this chapter we always denote by ‖ · ‖ the norm in L2. Let
E(t) = ∫ Ω (|∇u|2 − 1
2 |u|4 + |u|2)dx, (5.1.3)
(5.1.1) by ut, and integrating over Ω yields
1 2 dE
dt + ‖ut‖2 = 0. (5.1.4)
This implies that
E(t) ≤ E(0) = ∫ Ω (|∇φ|2 − 1
2 |φ|4 + |φ|2)dx. (5.1.5)
We infer from (5.1.2) that
1 2 d
dt ‖u‖2 + E(t)− 1
∫ Ω |u|4dx = 0. (5.1.6)
If E(0) < 0, then
1 2 d
dt ‖u‖2 ≥ 1
By the Cauchy-Schwartz inequality, we get∫ Ω |u|2dx ≤
where |Ω| denotes the measure of Ω. Let
E0(t) = ∫ Ω |u|2dx. (5.1.9)
Thus, combining (5.1.8) with (5.1.7) yields
dE0(t) dt
≥ C1E20 (5.1.10) with C1 being a positive constant depending only on Ω. Then we can get
E0(t) ≥ E0(0)1− C1E0(0)t (5.1.11)
by solving differential inequality (5.1.10). This implies that when t→ t0 with
t0 = 1
C1E0(0) , (5.1.12)
the solution u must blow up. Notice that the condition E(0) < 0 means that the initial data are sufficiently large. This can be seen from the following investigations. Consider that φ(x) = ku0(x) with u0(x) being given and k being positive constant. Then it follows from (5.1.5) that
E(0) = ∫ Ω (k2|∇u0|2 − 12k
4|u0|4 + k2|u0|2)dx (5.1.13)
must be negative. On the other hand, when initial data are small, as a consequence of the result in Section 3 of this chapter, problem (5.1.1) admits a unique global solution. This simple example indicates the need for considering global existence of solution to nonlinear evolution equations with small initial data. Investigations on the global existence with small initial data are not
new topics. For instance, as early as in 1948, R. Bellman [27] considered global existence of the solution to a semilinear parabolic equation with small initial data. However, from an unpublished manuscript by T. Nishida in 1975, a systematic approach, which combines local existence and uniqueness result with uniform a priori estimates, has been developed in a series of papers. Since the middle of the 1960s, some interesting results on the Cauchy problems for some special nonlinear heat equations and nonlinear wave equations have emerged. For instance, in the papers [55], [56] by H. Fujita, the author considered the following initial value problem:
ut −∆u = uα+1, (x, t) ∈ Rn ×R+,
u|t=0 = ϕ(x), x ∈ Rn, (5.1.14)
and proved that if α < 2n, then for any smooth initial data ϕ ≥ 0, ϕ 6≡ 0, the solution u must blow up in finite time no matter how small ϕ is. On the other hand, he also proved global existence if α > 2n and initial data are small. Later on, the blow-up results for the case α = 2n were obtained by K. Kobayashi et al. in [84] and by F.B. Weissler in [153], respectively. It is interesting to ask why global existence or nonexistence is related to the space dimension n. Does this type of relation between n and global existence pursue general fully nonlinear heat equations with small initial data? It has been discovered that for the Cauchy problem, solutions to linear evolution equations including a linear heat equation, a linear wave equation or a linear Schro¨dinger equation decay at a rate depending on the space dimension n. For instance, for the Cauchy problem of linear heat equation,
ut −∆u = 0, (x, t) ∈ Rn ×R+,
u|t=0 = ϕ(x), (5.1.15)
on the dimension n, namely, the L∞ norm and the L2 norm of solution decays at the rate t−
2 , and t− n
4 , respectively, as can be seen in the next section of this chapter. This explains why global existence or non-existence of solution to the Cauchy problem for nonlinear heat equation with small initial data depends on the space dimension n. Similar results hold for other linear evolution equations such as the linear wave equation, the linear Schro¨dinger equation, etc. For instance, for the Cauchy problem of linear wave equation:
utt −∆u = 0, (x, t) ∈ Rn ×R+,
u|t=0 = 0, ut|t=0 = ϕ(x), (5.1.16)
it has been proved that although the L2 norm of solution u is conserved, the L∞ norm of solution u decays at a rate t−
n−1 2 provided that ϕ(x),
say, is a smooth function with compact support. Since the end of the 1970s, study of global existence or blow-up in finite time for nonlinear evolution equations with small initial data has become a hot topic. S. Klainerman in the 1980s obtained the sharp result concerning the global existence of a smooth small solution to the Cauchy problem for fully nonlinear wave equations, showing that if the nonlinear term is of second order, then when n > 3, solution globally exists. Early examples by F. John and others show that for a particular nonlinear wave equation in three dimensions the solution blows up in a finite time. This indicates that the result by S. Klainerman is sharp regarding the space dimension. The sharp result for fully nonlinear heat equations was obtained in the paper [166] by S. Zheng and Y. Chen, and [159] by S. Zheng; see also the paper [120] by G. Ponce. We also refer to the monographs [164] by S. Zheng. In this chapter, we will present the results on the Cauchy problem for fully nonlinear heat equations mainly based on [164]. For the initial boundary value problem with the domain Ω being un-
bounded, i.e., the complement of another bounded domain in Rn, it has been discovered that the situation is very similar to the corresponding Cauchy problem, i.e., global existence or nonexistence of a solution has exactly the same relationship with the space dimension n as for the Cauchy problem. However, if an initial boundary value problem for nonlinear evolu-
tion equations in Ω × R+ with Ω being a bounded domain in Rn is concerned, then the situation is very different from the Cauchy problem regarding global existence or nonexistence of a small solution. The
of to linearized equation generally does not depend on the space dimension. In contrast, the long time behavior of the solution to the linearized equation heavily depends on the first eigenvalue of the corresponding elliptic operator subject to certain boundary conditions. In this aspect, let us look at two simple examples. First, consider the initial Dirichlet boundary value problem for the heat equation:
∂u
∂t −∆u = 0, (x, t) ∈ Ω×R+,
u|Γ = 0,
u(0) = φ(x)
where Ω is a bounded domain in Rn with the smooth boundary Γ. Let Φi(x) (i = 1, · · ·) be the normalized eigenfunction corresponding to the i-th eigenvalue λi with distribution 0 < λ1 < λ2 ≤ · · · ≤ λn ≤ · · · → +∞ :
−∆Φi = λiΦi, x ∈ Ω,
Φi|Γ = 0. (5.1.18)
Then, it is easy to verify that the solution to problem (5.1.17) can be expressed by
u(x, t) = ∞∑ i=1
where
Ai = ∫ Ω φ(x)Φi(x)dx. (5.1.20)
Thus, as time goes to infinity, the solution u to problem (5.1.17) exponentially decays to zero, and the decay rate does not depend on the space dimension n. As will be shown in the second section of this chapter, the nonlinear heat equation with a nonlinear term being at least a second-order perturbation, a global small smooth solution exists for any space dimension n.
