• 대한수학회논문집
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• 제23권2호
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• pp.211-227
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• 2008

As a mathematical model proposed to understand the behaviors of interacting species, cross-diffusion systems with functional responses of prey-predator type are considered. In order to obtain $W^{1_2}$-estimates of the solutions, we make use of several forms of calculus inequalities and embedding theorems. We consider the quasilinear parabolic systems with the cross-diffusion terms, and without the self-diffusion terms because of the simplicity of computations. As the main result we derive the uniform $W^{1_2}$-bound of the solutions and obtain the global existence in time.

• 대한수학회지
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• 제51권1호
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• pp.67-86
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• 2014

Based on an expanded mixed finite element method, we consider the semidiscrete approximations of the solution u of the quasilinear pseudo-parabolic equation defined on ${\Omega}{\subset}R^d$, $1{\leq}d{\leq}3$. We construct the semidiscrete approximations of ${\nabla}u$ and $a(u){\nabla}u+b(u){\nabla}u_t$ as well as u and prove the existence of the semidiscrete approximations. And also we prove the optimal convergence of ${\nabla}u$ and $a(u){\nabla}u+b(u){\nabla}u_t$ as well as u in $L^2$ normed space.