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Advances in Mathematical Applications Gakkotosho applied assume asymptotic Banach space blow-up boundary condition bounded chemotaxis compact consider continuous function convergence cosh Crum's theorem deduce defined Definition denote Dirichlet problem discrete variational disjoint domain effective components eigenvalues estimate family of triangulations finite difference finite difference scheme formulation Fujita problem G Xh given global existence Hence implies initial value inverse uniqueness property KdV equation Kusano Lemma linear Math Mathematical Sciences measures Moreover Navier-Stokes equations norm numerical obtain operator optimal oscillatory paper parabolic partial differential equations partitions Picone-type inequalities positive constants proof of Theorem Proposition prove regular Remark respectively result Riesz basis Sciences and Applications semigroup sequence sinh Sobolev space space Stefan problem Sturm-Liouville equation subdivision sufficient Theorem 4.1 Tokyo triangle triangulation Th unique solution vector field verify weak solutions zero