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2 2004

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On the wave equations with a dissipation anda source of cubic convolution type in Rn

Tatar N.-e.

Demonstratio Mathematica

2004

Vol. 37, nr 3

543--555

We study the interaction between a dissipative term and a source term of cubic convolution type for the wave equation in Rn. These terms have both the same form and involve convolutions with a singular kernel. The investigation will depend on the coefficient of the source term which is a functions of the time variable. Some results on the boundedness of the solutions are proved. Moreover, we establish an asymptotic stability result.

Some gradient estimates on covering manifolds

Dungey N.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

Vol. 52, nr 4

437-443

Let M be a complete Riemannian manifold which is a Galois covering, that is, M is periodic under the action of a discrete group G of isometries. Assuming that G has polynomial volume growth, we provide a new proof of Gaussian upper bounds for the gradient of the heat kernel of the Laplace operator on M. Our method also yields a control on the gradient in case G does not have polynomial growth.