Global existence and blow-up phenomena for p-Laplacian heat equation with inhomogeneous Neumann boundary conditions
© Li and Li; licensee Springer 2014
Received: 28 June 2014
Accepted: 15 September 2014
Published: 2 October 2014
In this paper, we consider a p-Laplacian heat equation with inhomogeneous Neumann boundary condition. We establish respectively the conditions on the nonlinearities to guarantee that the solution exists globally or blows up at some finite time. If blow-up occurs, we obtain upper and lower bounds of the blow-up time by differential inequalities.
MSC: 35K55, 35K60.
where is a real number, div denotes the scalar divergence operator, Ω is a bounded star-sharped region of () with smooth boundary ∂ Ω, n is the unit outward normal on ∂ Ω, is the outward normal derivative of u on ∂ Ω and is the blow-up time if blow-up occurs, or else .
The blow-up phenomena of solutions to various nonlinear problems, particularly for hyperbolic and parabolic systems, have received considerable attention in the recent literature. For work in this area, the reader can refer to –. Other contributions in the field can be found in – and the references cited therein. A variety of methods have been used to determine the blow-up of solutions and to indicate an upper bound for the blow-up time. To our knowledge, the first work on lower bound for was shown by Weissler , , but during the past several years a number of papers deriving lower bound for in various problems have appeared (see ).
where Ω is a bounded smooth domain in , , , , and studied the blow-up time, the location of the blow-up set, and the blow-up profile of the blow-up solution for sufficiently large D. In particular, they proved that for almost all initial data, if D is sufficiently large, then the solution blows up only near the maximum points of the orthogonal projection of the initial data from onto the second Neumann eigenspace.
and established respectively the conditions on nonlinearities to guarantee that exists globally or blows up at some finite time. If blow-up occurs, we obtain upper and lower bounds of the blow-up time.
and showed the backward uniqueness in time for solutions to Neumann and Dirichlet problems by energy methods and gave reasonable physical interpretation for the obtained conclusions.
Motivated by the above work, we intend to study the global existence and the blow-up phenomena for problem (1.1). It is well known that the data f and g may greatly affect the behavior of with the development of time. From the physical standpoint, −f is the heat source function, is the variational heat-conduction coefficient, is the heat-conduction function transmitting into interior of Ω from the boundary of Ω. We can deduce that if and , the blow-up phenomena of solution of (1.1) occur early under some conditions. Under the conditions that f and g are nonnegative functions, we can deduce that the solution of (1.1) is nonnegative and smooth. In this paper, by using differential inequalities, we establish the conditions on the nonlinearities to guarantee that exists globally or blows up at some finite time, respectively. If blow-up occurs, we obtain the upper and lower bounds of the blow-up time. The main innovational and novel points of this paper are: (a) the model is representative, for example, the model is the equation in , , – if with suitable f; (b) the problem considered in this paper is a nonlinear equation with inhomogeneous Neumann boundary dissipation, this problem is significant; (c) we give the reason and process of the definition of auxiliary functional; (d)since the model is general, the estimates are concise and precise.
The present work is organized as follows. In Section 2, we establish the conditions on the nonlinearities to guarantee that exists globally. In Section 3, we show the conditions on the nonlinearities which ensure that the solution blows up at some finite time and obtain the upper bound for the blow-up time. Section 4 is devoted to showing the lower bound of the blow-up time under some assumptions.
2 The global solution
In this section, we establish the conditions on the nonlinearities to guarantee that exists globally. We state our result as follows.
Thendoes not blow up, that is, exists for all time.
In order to prove this theorem, we give the following lemma.
Proof of Theorem 2.1
where . We note that in view of (2.3) and .
with , for small enough.
From (2.18), we can conclude that remains bounded for all time under the condition in Theorem 2.1. In fact, if blows up at finite time , then is unbounded near , which forces in some interval . So we have in , which implies that is bounded in , this is a contradiction.
The proof of Theorem 2.1 is completed. □
3 Blow-up and upper bound estimation of
In this section, we do not need Ω to be star-sharped. We establish the conditions to ensure that the solution of (1.1) blows up at finite time and derive an upper bound for the blow-up time . Now we state the result as follows.
whereis defined as (2.4). When (), we have.
which with imply for all .
If , we have (by ). Furthermore, by (3.6) we conclude that and is increasing for all . So .
The proof of Theorem 3.1 is completed. □
4 Lower bound estimation of
In this section, under the assumptions that is a bounded star-sharped domain and convex in two orthogonal directions, we establish a lower bound for the blow-up time . Now we state the result as follows.
In order to prove Theorem 4.1, we list the following lemmas.
The proof can be found in . □
This completes the proof. □
Proof of Theorem 4.1
for some positive constants , .
where and are computable positive constants.
The proof of Theorem 4.1 is completed. □
The authors would like to thank the referee for his/her careful reading and kind suggestions. Project supported by the National Natural Science Foundation of China (11201258), the Natural Science Foundation of Shandong Province of China (ZR2011AM008, ZR2011AQ006, ZR2012AM010), the Program for Scientific Research Innovation team in colleges and universities of Shandong Province.
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