Low regularity global solutions for nonlinear evolution equations of Kirchhoff type

We study the global solvability of the Cauchy–Dirichlet problem for two second order in time nonlinear integro-differential equations: 1)the extensible beam/plate equationutt+Δ2u−m(∫Ω|∇u|2dx)Δu=0(x∈Ω,t∈R);2)a special case of the Kirchhoff equationutt−(a+b∫Ω|∇u|2dx)−2Δu=0(x∈Ω,t∈R). By exploiting the I -method of J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, we prove that both equations admit global-in-time infinite energy solutions. In case 1), the energy is the mechanical energy; in case 2), it is a second order invariant introduced by S.I. Pokhozhaev. For the extensible beam equation 1), we also consider the effect of linear dissipation on such low regularity solutions, and we prove their exponential decay as t→+∞.