# Asymptotic behavior of solutions for a free boundary problem with a nonlinear gradient absorption

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## Abstract

This paper deals with the free boundary problem for a parabolic equation, \(u_t-u_{xx}=u^{p}-\lambda |u_x|^{q}\), \(t>0\), \(0<x<s(t)\), with \(p, q>1\). It is well known that global existence or blowup of solutions of nonlinear parabolic equations depends on which one dominating the model, the source or absorption, and on the absorption coefficient for the balance case of them. The aim of the paper is to study the influence of *p*, *q*, initial data and free boundary on the asymptotic behavior of solutions. At first, the ecological meaning of this model is explained by deriving the equation and the free boundary condition. Then, local existence and uniqueness are discussed, and the continuous dependence on initial data and comparison principle are proved. Furthermore, the finite time blowup and global solution are given by constructing sub- and super-solutions. In the case of \( p>q>1 \), our results show that the solution blows up in finite time for the initial value sufficiently large; but if the initial data are small, there exist global fast solutions which decay exponentially. Also, with a suitable initial value, the existence of global slow solution with at most polynomial decaying is established. While, in the case of \(q\ge p>1\), all solutions with nonnegative initial data of exponential decay exist globally. Finally, the problem with double free boundaries, \(u_tu_{xx}=u^p-\lambda |u_x|^q, t>0, g(t)<x<h(t)\), is also discussed and the similar conclusions are obtained.

## Mathematics Subject Classification

35A01 35B40 35B44 35K58## Notes

### Acknowledgements

We are grateful to Professor Bei Hu for his encouragement and helpful suggestions during the completion of this paper. Also, we would like to express our sincere gratitude to the referee for a very careful reading of the paper and for all his (or her) insightful comments and valuable suggestions. This work is supported by the National Natural Science Foundation of China (Nos. 11371286, 11401458).

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