• Shim Seong-A (Department of Mathematics Sungshin Women's University)
  • Published : 2006.04.01


Using calculus inequalities and embedding theorems in $R^1$, we establish $W^1_2$-estimates for the solutions of prey-predator population model with cross-diffusion and self-diffusion terms. Two cases are considered; (i) $d_1\;=\;d_2,\;{\alpha}_{12}\;=\;{\alpha}_{21}\;=\;0$, and (ii) $0\;<\;{\alpha}_{21}\;<\;8_{\alpha}_{11},\;0\;<\;{\alpha}_{12}\;<\;8_{\alpha}_{22}$. It is proved that solutions are bounded uniformly pointwise, and that the uniform bounds remain independent of the growth of the diffusion coefficient in the system. Also, convergence results are obtained when $t\;{\to}\;{\infty}$ via suitable Liapunov functionals.


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