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Global existence and blow-up phenomena for p-Laplacian heat equation with inhomogeneous Neumann boundary conditions
Boundary Value Problems volume 2014, Article number: 219 (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.
In this paper, we deal with the initial-boundary value problem
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 ).
Under suitable conditions on the nonlinearities, they determined a lower bound on the blow-up time when blow-up occurs. In addition, a sufficient condition which implies that blow-up does occur was determined. Ding and Guo  studied the global solution and blow-up solution of the equation
where is a bounded domain with smooth boundary ∂D. Under appropriate assumptions on the functions a, b, f, g and h, by constructing auxiliary functions and using maximum principles, the sufficient conditions for the existence of global solution or blow-up solution, an upper estimate of the global solution, an upper bound of the blow-up time and an upper estimate of the blow-up rate were specified. Mizoguchi  studied the semilinear heat equation
and showed that if u blows up at , then for some . Ishige and Yagisita  considered the blow-up problem for the semilinear heat equation
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.
In recent paper, Payne et al. considered
and established conditions on nonlinearities sufficient to guarantee that exists for all time or blows up at some finite time . Moreover, an upper bound for was derived. Under somewhat more restrictive conditions, a lower bound for was derived. Moreover, in , Payne et al. investigated
and showed that blow-up occurs at some finite time under certain conditions on the nonlinearities and the data, upper and lower bounds for the blow-up time were obtained when blow-up occurs. Li and Li  investigated
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.
Li  considered the p-Laplacian heat-conduction model
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.
Let u be a classical solution of (1.1), assume that the nonnegative functions f and g satisfy the following conditions:
where, , , and
Thendoes not blow up, that is, exists for all time.
In order to prove this theorem, we give the following lemma.
Let Ω be a bounded star-sharped region in, . Then, for any nonnegativefunction w and, we have
Proof of Theorem 2.1
By the divergence theorem and (2.2), we obtain
Applying Lemma 2.1, we have
Clearly, for all ,
Applying Hölder’s inequality and Young’s inequality, we get
Choose . Equation (2.11) implies
Using Hölder’s inequality and Young’s inequality, we get
where . We note that in view of (2.3) and .
with , for small enough.
Application of Hölder’s inequality leads to
Using Hölder’s inequality, we have
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.
Letbe the nonnegative solution of problem (1.1), and assume that the nonnegative integrable functions f and g satisfy the following conditions:
and θ satisfies
Moreover, we assumewith
Thenblows up at time, with
whereis defined as (2.4). When (), we have.
Using the divergence theorem and the assumptions on f, g, we obtain
which with imply for all .
Multiplying (3.3) by , we deduce
Integrating (3.4) over implies
If , (3.6) can be written as
Integrating (3.7) over , we obtain
which with (3.5) implies as . Therefore, for ,
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.
Letbe the nonnegative solution of problem (1.1), andblow up at time; moreover, the nonnegative functions f and g satisfy the following conditions:
for some constantsand. Define
Then satisfies the inequality
where, are computable (nonnegative) constants. It follows thatis bounded below by
In order to prove Theorem 4.1, we list the following lemmas.
Letbe a bounded star-sharped domain and convex in two orthogonal directions. Then, for any nonnegativefunction w and, we have
The proof can be found in . □
For alland, we have
Let . Since for all , we have
This completes the proof. □
Proof of Theorem 4.1
Differentiating , we obtain
By Lemma 2.1, we have
Firstly, we give the estimation of . Application of Lemma 4.1 leads to
Using Hölder’s inequality, we have
By Hölder’s inequality, we have
for some positive constants , .
Secondly, we estimate . Using Hölder’s inequality, we have
Application of Hölder’s inequality leads to
By Hölder’s inequality and Lemma 4.1, we get
Using Hölder’s inequality, we obtain
where and are computable positive constants.
Using Young’s inequality, we obtain
for . So we have the following inequalities:
Integrating (4.21) over , we get
As blows up, letting , we get the bound for as follows:
The proof of Theorem 4.1 is completed. □
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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.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.