## Advances in Mathematical Sciences and Applications, Volume 11, Pages 483-891 |

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absorbing set Advances in Mathematical Appl Applications Gakkotosho apply assume assumptions asymptotic attractor belongs boundary conditions boundary value problem bounded Cauchy problem compact consider continuous function contraction convergence Corollary defined definition denote depending derive differential equations differential inequality direct summand Dirichlet forms Dirichlet problems dissipation domain elliptic equations finite free boundary problems Hence Hilbert space holds implies inequality initial data inner product integrate introduce isometry Kenmochi limsup linear Lipschitz continuous Lq decay estimates Math Mathematical Sciences method monotone operator Moreover nonlinear nonnegative nonoscillatory solution norm obtain ODE system oscillation oscillatory paper phase positive constant proof of Theorem property PF Proposition prove regularity Remark respect right-hand side satisfies condition satisfying 2.1 Sciences and Applications semigroup sequence solution u satisfying space structure of solutions Suppose term Theorem 3.1 Type uniformly zero