Abstract
We study a nonlinear diffusion equation of the form \(u_t=u_{xx}+f(u)\ (x\in [g(t),h(t)])\) with free boundary conditions \(g'(t)=-u_x(t,g(t))+\alpha \) and \(h'(t)=-u_x(t,h(t))-\alpha \) for some \(\alpha >0\). Such problems may be used to describe the spreading of a biological or chemical species, with the free boundaries representing the expanding fronts. When \(\alpha =0\), the problem was recently investigated by Du and Lin (SIAM J Math Anal 42:377–405, 2010) and Du and Lou (J Euro Math Soc arXiv:1301.5373). In this paper we consider the case \(\alpha >0\). In this case shrinking (i.e. \(h(t)-g(t)\rightarrow 0\)) may happen, which is quite different from the case \(\alpha =0\). Moreover, we show that, under certain conditions on \(f\), shrinking is equivalent to vanishing (i.e. \(u\rightarrow 0\)), both of them happen as \(t\) tends to some finite time. On the other hand, every bounded and positive time-global solution converges to a nonzero stationary solution as \(t\rightarrow \infty \). As applications, we consider monostable, bistable and combustion types of nonlinearities, and obtain a complete description on the asymptotic behavior of the solutions.
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Acknowledgments
The authors would like to thank Professors Y. Du and Z. Lin for valuable discussion on the free boundary conditions.
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This research was partly supported by NSFC (No. 11271285) and by Shanghai Natural Science Foundation (No. 13ZR1454900).
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Cai, J., Lou, B. & Zhou, M. Asymptotic Behavior of Solutions of a Reaction Diffusion Equation with Free Boundary Conditions. J Dyn Diff Equat 26, 1007–1028 (2014). https://doi.org/10.1007/s10884-014-9404-z
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DOI: https://doi.org/10.1007/s10884-014-9404-z