Global existence, nonexistence and asymptotic behavior of solutions for the Cauchy problem of semilinear heat equations☆
Introduction
On the global existence and blowup of solutions for the Cauchy problem of the semilinear heat equation there have been many results [1], [2], [4], [5], [6], [7], [8], [9], [10], [12], [13], [14], [18], [19], [20], [21]. In 1992, Wang and Ding [17] studied the Cauchy problem where if ; if . Letting be the steady state solution of problem (1.1), (1.2), they proved that
- (i)
If , then problem (1.1), (1.2) admits a global solution.
- (ii)
If and , then the solution as .
- (iii)
If and , then the solution blows up in finite time.
In [3], the Cauchy problem was studied, and similar results were derived.
In [11] problem (1.1), (1.2), for the general case not satisfying or , was studied again. And some sufficient conditions for the global existence and nonexistence of solutions were given.
Throughout the present paper, the following notations are used for precise statements: denotes the usual space of all -functions on with norm and ; denotes the usual Sobolev space on with norm .
In the present paper, we consider the Cauchy problem where satisfies
- (H)
if ; if .
First, a family of potential wells and its corresponding sets are introduced. Then, we give a series of their properties. By using them, we not only prove the invariance of some sets under the flow of (1.3), (1.4), but also get isolate solutions. Further, we obtain a threshold result for the global existence and nonexistence of solutions. Finally, the asymptotic behavior of solutions is discussed.
Although there has been a lot of work using the potential well method, almost all of this is on the initial boundary value problem on a bounded domain. And there are few results dealing with the Cauchy problem. In [15], Nakao and Ono studied the Cauchy problem of the semilinear wave equation by using the potential well method, but they required the initial function to possess a bounded support, which means this problem can be resolved by using the potential well method on a bounded domain. In particular, for the global existence, finite time blowup and asymptotic behavior of solutions for the Cauchy problem of nonlinear parabolic equations with critical initial conditions, to our knowledge, there is no any result up to now. In this paper we obtain the corresponding threshold results on the global existence, nonexistence and asymptotic behavior of solutions of problem (1.3), (1.4) under some conditions.
We define some functionals and sets as follows where Then the main results obtained in this paper can be described as follows.
Assume that satisfies (H), and , where is the depth of potential well . Then
- (i)
Problem (1.3), (1.4) admits a global weak solution with and for , provided .
- (ii)
The global weak solution of problem (1.3), (1.4) decays to zero exponentially as , provided .
- (iii)
The weak solution of problem (1.3), (1.4) blows up in finite time, provided .
Section snippets
Preliminary lemmas and introducing of the family of potential wells
In this section, we shall introduce a family of potential wells and its corresponding sets , and give a series of their properties for problem (1.3), (1.4). And we always assume that satisfies (H). First, let the definitions of functionals , and the potential well with its depth given above hold. Then we define the outside of the potential well by
Next, we give some properties of above sets and functionals as follows.
Lemma 2.1 Let , . Then
Invariant sets and isolating of solutions
In this section, we prove the invariance of some sets under the flow of (1.3), (1.4), and isolate solutions for problem (1.3), (1.4). For this purpose, we give the definition of the weak solution first as follows.
Definition 3.1 A function with , is called a weak solution of problem (1.3), (1.4) on if the following conditions are satisfied in ;
Definition 3.2 Let be a weak
Global existence and finite time blowup of solutions
In this section, we establish the global existence (in time) and finite time blowup of solutions, and then give a threshold result for the global existence and nonexistence of solutions for problem (1.3), (1.4).
Definition 4.1 Let be a weak solution of problem (1.3), (1.4). We call a blowup in finite time if the maximal existence time is finite and
Theorem 4.1 Let satisfy (H), . Assume that , . Then problem(1.3),(1.4)admits a global weak solution
Problem (1.3), (1.4) with critical initial condition
In this section, we prove the global existence and nonexistence of solutions for problem (1.3), (1.4) with the critical initial condition .
Theorem 5.1 Let satisfy (H), . Assume that and . Then problem(1.3),(1.4)admits a global weak solution with and for .
Proof First implies that . Take a sequence such that , and as . Let . Consider the initial conditions
Asymptotic behavior of solutions
In this section, we discuss the asymptotic behavior of solutions for problem (1.3), (1.4).
Theorem 6.1 Let satisfy (H), . Assume and . Then, for the weak global solution of problem(1.3),(1.4), there exists a constant such that
Proof First, Theorem 4.1 gives the existence of global weak solutions for problem (1.3), (1.4). Now we only need to prove (6.1). Let be any global weak solution of problem (1.3), (1.4) with , . Then (3.1) holds for
Acknowledgement
We are very thankful for the reviewer’s careful reading of our manuscript and the constructive and advisable recommendations. The referee gave a lot of valuable suggestions, which made this work much better and is also helping us in our further research. We also thank Gao Yang for his kindly help.
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This work is supported by National Natural Science Foundation of China (10271034), National Natural Science Foundation of Heilongjiang Province, Harbin Engineering University Foundational Research Foundation (HEUF04012).