Well-posedness results and blow-up for a class of semilinear heat equations

This paper considers the initial value problem for nonlinear heat equation in the whole space RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{N}$\end{document}. The local existence theory related to the finite time blow-up is also obtained for the problem with nonlinearity source (like polynomial types).


Introduction
We consider the following initial value problem for u : where λ, ρ ∈ R and m, N ∈ N * are parameters; is the standard Laplacian with Dirichlet boundary conditions in L 2 (R N ); u = u(x, t) is the state of the unknown function and f is given function. Mathematical formulations modeled by problem (P) appear in many other practical applications of mathematics and engineering science models [5][6][7]12]. The goal of this paper is the study of the local existence, unique continuation and a finite time blowup of solution to Problem (P). For the homogeneous linear case of problem (P) this is the well-known classical heat equation and these problems have been studied for decades and much work on this topic has been published.
In recent years, results on inhomogeneous partial differential equations have been extensively studied. Unlike linear source functions, nonlinear source functions describe more complex systems and have many applications in the real world. For nonlinear problems that appear in some physical phenomena there are many results devoted to nonlinear heat equations; see [2][3][4][5][6][7][8][9][10][11]13] and the references therein. Studies of well-posedness and asymptotic behavior of solutions for semilinear heat equations have been performed by many authors; see [1,7,14] and the references therein. Equations of the form on a bounded one-dimensional domain were studied by Chafee and Infante (1974) (so this equation is sometimes called the Chafee-Infante equation). Although there has been published much work on the semilinear cases, the literature on the case of the form of Problem (P) is quite scarce. In this work, we consider the special case with . Moreover, we also establish the results on the continuation and finite time blow-up of solutions in H 2s (R N ). We assume that the following hypotheses hold: for C > 0, m > 1.

Functional setting and notation
The notation · B stands for the norm in the Banach space B. For 1 ≤ p ≤ ∞, T > 0, consider the Banach space of real-valued measurable functions f : (0, T) → B with the following norm: For s ∈ R N , the Sobolev space H s (R N ) consists of all tempered distributions w ∈ S (R N ) whose Fourier transform w is a regular distribution such that The inner product and norm of w, v ∈ H s (R N ) are defined by Let A denote the negative Laplacian operator in L 2 (R N ), We define A as an operator acting in L 2 (R N ) because we can study it explicitly by use of the Fourier transform. As is well known, A is a closed, densely defined positive operator, and -A is the generator of a strongly continuous contraction semigroup The Fourier representation of the semigroup operators is If t > 0 we have for any s > 0 We define the nonlinear function

Some related results
We define mild H 2s (R N )-value solutions of (P) as follows.
such that where e -tA is given by (2.5), and φ m is given by (1.2).

Lemma 2.2
Let e -tA be the semigroup operator defined in (2.5) and s > 0. If t > 0, then Proof Suppose that w ∈ L 2 (R N ). Using the Fourier representation (2.5) of e -tA as multiplication by e -t|ξ | 2 and the definition of the H 2s -norm, we get Hence, by Parseval's theorem, we have z s e -tz ≤ Ce t t s , (2.9) and the result of this lemma follows.

Local existence of the solutions to problem (P)
Evaluating the τ -integral, with s < 1, and taking the supremum of the result over 0 ≤ t ≤ T, we get For s ≥ N 4 , we have the Sobolev embeddings and using the facts that where 0 < ρ < 1. Moreover, in that case The contraction mapping theorem then implies the existence of a unique solution u ∈ B R .

Uniqueness continuation of the solution
Since we already know that the mild solution of Problem (P) does exist, the question is whether it will continue (continuation to a bigger interval of existence) and in what situation it is non-continuation by blow-up.

Definition 3.2 Given a mild solution u ∈ C([0, T]
; H 2 (R N )) of (P), we say that u is a con- where Q T,R = R + u(·, T) H 2s (R n ) . For T ≥ T > 0 and R > 0, let us define (3.5) • Step I: We show that J defined as in (3.1) is the operator on B R . Let u ∈ B R and we consider two cases. Case 1: If t ∈ [0, T], then by virtue of Theorem 3.1, we have the Problem (P) has a unique solution and we also have u (·, t) = u(·, t). Thus Ju (t) = Ju(t) = u(·, t) for all t ∈ [0, T].
Case 2: If t ∈ [T, T ], we have Estimating the term (I), using (2.9) we have, for all t ∈ [T, T ], Hence, we get From (3.4a), this implies that the following estimate holds: From Lemma 2.2, ( H 2 ), we have the following estimate for all t ∈ [T, T ]: Using (3.4b), we obtain From (2.9) and ( H 2 ), we have the following estimate for all t ∈ [T, T ]: Using (3.4c), we infer that It follows from (3.7), (3.8), (3.10) that, for every t ∈ [0, T ] We have shown that J is a map B R into B R . • Step II: We show that J is a contraction on B R . Let u, v ∈ B R , and we have, for 0 ≤ t ≤ T , So without loss of generality, we may assume that 0 ≤ R < 1 4 , this implies that J is a R 4contraction. By the Banach contraction principle it follows that J has a unique fixed point u of J in B R , which is a continuation of u. This finishes the proof.

Finite time blow-up
The next results are on global existence or non-continuation by a blow-up. u(x, t) be a solution of (P). We define the maximal existence time T max of u(x, t) as follows:
(ii) If there exists T ∈ [0, ∞) such that u(x, t) exists for 0 ≤ t < T, but does not exist at t = T, then T max = T. Assume that T max < ∞, and u(·, t) H 2s (R N ) ≤ R 0 , for some R 0 > 0. Now suppose there exists a sequence {t k } k∈N ⊂ [0, T max ) such that t k → T max and {u(·, t k )} k∈N ⊂ H 2s (R N ). Let us show that {u(·, t k )} k∈N is a Cauchy sequence in H 2s (R N ). Indeed, given > 0, fix N ∈ N such that, for all k, n > N , 0 < t k < t n < T max , we have u(·, t n )u(·, t k ) = e -(t n -t k )A -I e -t k A f + ≤ e -(t n -t k )A -I L(H 2s (R N )) u(·, t k ) H 2s (R N ) . (3.15)