Global and blow-up solutions for quasilinear parabolic equations with a gradient term and nonlinear boundary flux
© Li et al.; licensee Springer. 2014
Received: 27 February 2014
Accepted: 29 May 2014
Published: 5 June 2014
This work is concerned with positive classical solutions for a quasilinear parabolic equation with a gradient term and nonlinear boundary flux. We find sufficient conditions for the existence of global and blow-up solutions. Moreover, an upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’ and an upper estimate of the global solution are given. Finally, some application examples are presented.
MSC:35R45, 35K65, 34A12.
Here () is a bounded domain with a smooth boundary ∂D, is the closure of D, , n is the outer normal vector and T is the maximum existence time of . , and are nonlinear diffusion coefficient, reaction term and boundary flux, respectively. Let , , and suppose that the function , for any , , , , is a nonnegative function, , is a positive function, and the positive function satisfies the compatibility conditions. Under these assumptions, the classical parabolic equation theory [, Section 3] ensures that there exists a unique classical solution to problem (1.1)-(1.3) for some , and the solution is positive over . Moreover, by the regularity theorem [, Chapter 3], we know .
Equation (1.1) describes the diffusion of concentration of some Newtonian fluids through porous media or the density of some biological species in many physical phenomena and combustion theories (see [3, 4]). The nonlinear Neumann boundary value condition (1.2) can be physically interpreted as the nonlinear radial law (see, e.g., [5, 6]).
with the homogeneous Dirichlet boundary condition. By estimating the integral of ratio of one solution to the other, the author proved both global existence and blow-up results. Then he used the same method to study a more generalized equation with a gradient term, see .
subject to boundary flux , and they obtained sufficient conditions for the global existence and blow-up solution, the upper estimate of global solution and blow-up time.
Motivated by the above works, we construct an appropriate auxiliary function and use the Hopf maximum principle to study problem (1.1)-(1.3). The aim of this paper is to obtain sufficient conditions for the existence of blow-up and global solution, an upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’ and an upper estimate of the global solution and then to give some examples.
2 Main results and proof
We now state and prove the main results of this paper. Firstly, we give sufficient conditions of the existence of a blow-up solution of problem (1.1)-(1.3).
- (1)For any ,(2.1)
- (2)For any ,(2.2)(2.3)
- (3)For any ,(2.4)
- (4)The constant(2.5)
- (5)The integration(2.6)
where , , and is the inverse function of Φ.
The proof is completed. □
The result on the global solution is stated as Theorem 2 below.
- (1)For any ,(2.26)
- (2)For any ,(2.27)(2.28)
- (3)For any ,(2.29)
- (4)The constant(2.30)
- (5)The integration(2.31)
where , , and is the inverse function of Ψ.
The proof is completed. □
In what follows, we present several examples to demonstrate the applications of Theorems 1 and 2.
This work is supported by the Natural Science Foundation of Shandong Province of China (ZR2012AM018) and the Fundamental Research Funds for the Central Universities (No. 201362032). The authors would like to deeply thank all the reviewers for their insightful and constructive comments.
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