The author examined non-zero -periodic (in time) solutions for a semilinear beam equation under the condition that the period is an irrational multiple of the length. It is shown that for a.e. (in the sense of the Lebesgue measure on ) the solutions do exist provided the right-hand side of the equation is sublinear.
For a nonlinear hyperbolic equation defined in a thin domain we prove the existence of a periodic solution with respect to time both in the non-autonomous and autonomous cases. The methods employed are a combination of those developed by J. K. Hale and G. Raugel and the theory of the topological degree.
We consider on a two-dimensional flat torus defined by a rectangular periodic cell the following equationIt is well-known that the associated energy functional admits a minimizer for each . The present paper shows that these minimizers depend actually only on one variable. As a consequence, setting to be the first eigenvalue of the Laplacian on the torus, the minimizers are identically zero whenever . Our results hold more generally for solutions that are Steiner symmetric, up to a translation....
We show that the periodic Camassa–Holm equation possesses a global continuous semigroup of weak conservative solutions for initial data in . The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measure with . The total energy is preserved by the solution.
Let Ω ⊂ RN be a smooth bounded domain. We give sufficient conditions (which are also necessary in many cases) on two nonnegative functions a, b that are possibly discontinuous and unbounded for the existence of nonnegative solutions for semilinear Dirichlet periodic parabolic problems of the form Lu = λa (x, t) up - b (x, t) uq in Ω × R, where 0 < p, q < 1 and λ > 0. In some cases we also show the existence of solutions uλ in the interior of the positive cone and that uλ can...