The Global Solution Of A Kind Of Semi-linear Wave Equation With Time Decay And Dissipation Coefficient

In this paper,we study the semi-linear wave equation with time-dependent dissipation as follows:(?)where t ∈[0,∞),α<1,μ>1.We consider the global existence of small data solution.In the process of proof,we use the Duhamel’s principle to express the solution of the semi-linear wave equation.It requires us to consider the following parameter-dependent Cauchy problem for the damped wave equation:(?)where t ∈[0,∞),α<1,μ>1.In this paper,we can obtain the decay estimates of ||v(t,·)||L2,||▽v(t,·)||L2 and||(?)v(t,·)||L2.Using these estimates,with the condition of small data.and Con-tractible mapping principle,we can have the global solution of the problem(0.1).We get the specific conclusions as follows:There is a positive constant ε0,assuming that||u1||H1 + ||u2||L2 ≤ε for ε≤ε0.and(?)the Cauchy problem(0.1)has a unique global solution u∈ C([0,∞),H1)∩ C1([0,∞),L2).