Applications of Fixed Point and Approximate AlgorithmsView this Special Issue
The Global Existence of Nonlinear Evolutionary Equation with Small Delay
We investigate the global existence of the delayed nonlinear evolutionary equation . Our work space is the fractional powers space . Under the fundamental theorem on sectorial operators, we make use of the fixed-point principle to prove the local existence and uniqueness theorem. Then, the global existence is obtained by Gronwall’s inequality.
On the existence for solutions of evolutionary equations, there are many works and methods [1–7]. For example, the fixed principle [1, 3–5, 7] and Galerkin approximations [2, 6]. They are very classical methods to prove existence and uniqueness. Generally speaking, there are four solution concepts. That is, weak solution, mild solution, strong solution, and classical solution. We can obtain different types for different conditions. For instance , consider the following inhomogeneous initial value problem: where is Banach space. If the nonlinearity , the initial value problem has a unique mild solution. If the nonlinearity is differentiable a.e. on and , then for every the initial value problem has a unique strong solution. Furthermore, if the nonlinearity is locally Hölder continuous, then the initial value problem has a unique classical solution.
In the article , the author considered scalar reaction-diffusion equations with small delay There the nonlinearity is assumed to be locally lipschitz and to satisfy the one-sided growth estimates for some continuous . To prove existence, he treated the equation stepwise as a nonautonomous undelayed parabolic partial differential equation on the time intervals by regarding the delayed values as fixed. His strategy was to mimic the results of Henry [3, Theorem 3.3.3 and Corollary 3.3.5], but with his assumption of Hölder continuity in replaced by -integrability. Many authors had investigated the nondelayed one in [8–10].
In this paper, we consider the following nonlinear evolutionary equation with small delay: Under the hypothesis of (A1), (A2), and (A3) (see Section 2), we firstly make use of the fixed principle to prove the local existence and uniqueness theorem. Then we obtain the global existence and uniqueness by Gronwall inequality. In the whole paper, our work space is fractional powers space . Its definition can be referred to [1, 3, 4].
In this section, we will give some basic notions and facts. Firstly, basic assumptions are listed.(A1) Let be a positive, sectorial operator on a Banach Space . is an analytic semigroup generated by . Fractional powers operator is well defined. Fractional powers space with the graph norm . For simplicity, we will denote as .(A2) For some , the nonlinearity is locally Lipschitz in . More precisely, there exists a neighborhood such that for and some constant (A3) The initial value is Hölder continuous from to .
Obviously the (classical) solution of (1.4) can be expressed by the variation of constant formula where we let . Next we come to the main theorem on analytic semigroup which is extremely important in the study of the dynamics of nonlinear evolutionary equations .
Theorem 2.2 (fundamental theorem on sectorial operators). Let be a positive, sectorial operator on a Banach Space and be the analytic semigroup generated by . Then the following statements hold.(i)For any , there is a constant such that for all (ii)For , there is a constant such that for and (iii) For every , there is a constant such that for all and
3. Main Results
Theorem 3.1. Suppose (A1), (A2), and (A3) hold. Then there exists a sufficiently small such that (1.4) has a unique solution on .
Proof. For convenience, we still denote . Select and construct set
Let , choose sufficient small such that
Let be the Banach space with the usual supremum norm which we denote by . Let be the nonempty closed and bounded subset of defined by
On we define a mapping by
Next we will utilize the contraction mapping theorem to prove the existence of fixed point. In order to complete this work, we need to verify that maps into itself and is a contraction mapping on with the contraction constant ≤1/2.
It is easy to see from (3.4) and (3.5) that for , . For , considering (2.1), (2.3), (3.2), and (3.3), we obtain Therefore . Furthermore if then from (3.3) and (3.5) which implies that By the contraction mapping theorem the mapping has a unique fixed point . This fixed point satisfies the following:
From (2.1) and the continuity of it follows that is continuous on and a fortiori bounded on this interval. Let Next we want to show that is locally Hölder continuous on . To this end, we show first that the solution of (3.9) is locally Hölder continuous on .
Select , such that Considering (2.3) and (2.4), we select such that Synthesizing (3.12) and (3.13), we get
So we proved the solution of (3.9) is locally Hölder continuous on . Furthermore, in view of (2.1) we have
Let be the solution of (3.9) and (3.10) and . In view of locally Hölder continuous on of , consider the inhomogeneous initial value problem By Corollary 4.3.3 in , this problem has a unique solution and the solution is given by Each term of (3.17) is in and a fortiori in . Operating on both sides of (3.17) with we find By (3.9) the right-hand side of (3.18) equals and therefore . So for , by (3.17) we have So is a solution of (1.4). The uniqueness of follows readily from the uniqueness of the solutions of (3.9) and (3.16), and the proof is complete.
Before giving our global existence theorem, we should first prove extended theorem of solution.
Theorem 3.2 (extended theorem). Assume that (A1), (A2), and (A3) hold. And also assume that for every closed bounded set , the image is bounded in . If is a solution of (1.1) on , then either or there exists a sequence as such that . (If is unbounded, the point at infinity is included in .)
Proof. Suppose , there exists a closed bounded subset of and such that for . We prove there exists such that
in , which implies the solution may be extended beyond time .
Now let We show firstly that remains bounded as for any .
Observe that if , , in view of (2.3) and (3.19) we have
Secondly, suppose , so From (2.3) and (2.4) we get Thus (3.20) holds, and the proof is completed.
Theorem 3.3 (global existence and uniqueness). Assume that (A1), (A2), and (A3) hold. And for all , satisfies Then, the unique solution of (1.4) exists for all .
Proof. We need to verify that is bounded when . As for
Considering (3.25), we can obtain
Case 1. If . Because for and is Hölder continuous from to . Let From (3.27), we immediately get From Lemma 2.3, that is, Gronwall’s inequality, we find .
Case 2. If , still let because From (3.27) again, we obtain By Gronwall’s inequality again, we get . This completes the proof of this theorem.
This work is supported by NNSF of China (11071185) and NSF of Tianjin (09JCYBJC01800).
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