Global existence and uniqueness of the solution to a nonlinear parabolic equation

Consider the equation $$ u'(t)-\Delta u+|u|^\rho u=0, \quad u(0)=u_0(x), (1), $$ where $ u':=\frac {du}{dt}$, $ \rho=const>0, $ $x\in \mathbb{R}^3$, $t>0$. Assume that $u_0$ is a smooth and decaying function, $$\|u_0\|\:=\sup_{x\in \mathbb{R}^3, t\in \mathbb{R}_+} |u(x,t)|.$$ It is proved that problem (1) has a unique global solution and this solution satisfies the following estimate $$\|u(x,t)\|0$ does not depend on $x,t$.

The novel points in this work are: a) There is no restriction on the upper bound of ρ.
In [1], (Section 1.1) a nonlinear hyperbolic equation with the same nonlinearity is studied in a bounded domain, uniqueness of the solution is proved only for ρ ≤ 2/(n − 2), and existence is proved by a different method. The contraction mapping theorem is not used.
In [2] the quasi-linear problems for parabolic equations are studied in Chapter 5 in a bounded domain and under the assumptions different from ours. There are many papers and books on non-linear problems for parabolic equations (see the bibliography in [1], [2]. b) Existence of the global solution is proved. c) Method of the proof differs from the methods in the cited literature. Our result is formulated in Theorem 1: Theorem 1. Problem (1) has a unique global solution in X for any u 0 ∈ X.

Proofs
Let g(x, t) = e −|x| 2 (4πt) 3/2 . If u solves (1) then where := R 3 . Let X be the Banach space of continuous in R 3 ×R + functions, The Q is a contraction on B(R) if then Q is a contraction in B(R) in the Banach space X T with the norm · , t ∈ [0, T ]. We use the same notations for the norms in X T and in X ∞ .
We have proved that For T satisfying (5)-(6) there exists and is unique the solution to (1), and this solution can be obtained from (3) by iterations.
The problem now is: Does this solution exist and is unique on R + ? From our proof it follows that if the solution exists and is unique in X T , then the solution exists and is unique in X T 1 for some T 1 > T .
To prove that the solution u(x, t) to (1) exists on R + , assume the contrary: this solution does not exist on any interval [0, T 1 ), T 1 > T , where T is the maximal interval of the existence of the continuous solution. Then lim t→T −0 u(x, t) = ∞, because otherwise there is a sequence t n → T − 0 such that u(x, t n ) → u(x, T ) and one may construct the solution defined on [T, T 1 ], T 1 > T , by using the local existence and uniqueness of the solution to (1) with the initial value u(x, T ) for t ∈ [T, T 1 ]. This contradicts the assumption that T is the maximal interval of the existence of the continuous solution u.
Thus, if T < ∞ then one has lim t→T −0 u(x, t) = ∞. Let us prove that this also leads to a contradiction. Then we have to conclude that T = ∞ and Theorem 1 is proved.
We need some estimates. Multiply (1) by u, integrate over R 3 with respect to x, and then integrate by parts the second term. The result is: where N(u) := u 2 dy. Integrate (7) with respect to time over [0, T ] and get 0.5N(u(T )) + T 0 N(gradu) + |u| ρ+2 dy dτ = 0.5N(u(0)). Therefore, If (10) is proved then T is not the maximal interval of the existence of the solution to (1). This contradiction proves Theorem 1.
Proof of Lemma 1. One uses the Hölder inequality twice and gets By the last inequality (9) it follows that T 0 dτ |u| ρ+2 dy < c ∀T > 0, where c > 0 is a constant independent of T . The last integral in (11) is also bounded independently of T . It can be calculated analytically.
✷ The ideas related to the ones used in this paper were developed and used in [3]- [5].