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Note on blow-up of solutions for a porous medium equation with convection and boundary flux

Zhiyong Wang;  Jingxue Yin — 2012

Colloquium Mathematicae

De Pablo et al. [Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), 513-530] considered a nonlinear boundary value problem for a porous medium equation with a convection term, and they classified exponents of nonlinearities which lead either to the global-in-time existence of solutions or to a blow-up of solutions. In their analysis they left open the case of a certain critical range of exponents. The purpose of this note is to fill this gap.

Periodic solutions for a class of non-autonomous Hamiltonian systems with p ( t ) -Laplacian

Zhiyong Wang;  Zhengya Qian — 2024

Mathematica Bohemica

We investigate the existence of infinitely many periodic solutions for the p ( t ) -Laplacian Hamiltonian systems. By virtue of several auxiliary functions, we obtain a series of new super- p + growth and asymptotic- p + growth conditions. Using the minimax methods in critical point theory, some multiplicity theorems are established, which unify and generalize some known results in the literature. Meanwhile, we also present an example to illustrate our main results are new even in the case p ( t ) ≡ p = 2 .

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