Invariant Sets And Exact Solutions To Higher-dimensional Reaction-diffusion Equations And Wave Equations

The nonlinear phenomena widely appear in almost all the scientific fields such as physics, chemistry, biology, society, economy and so on. With the development of science, researches on the nonlinear systems are making more and more progress. Up to date, there is not a general method to solve the nonlinear PDEs. In past decades, a lot of methods have been established and developed to solve the nonlinear PDEs.The idea of this paper is to use invariant sets to solve higher-dimensional reaction-diffusion equations and higher-dimensional wave equations. Firstly, we study (3+1)-dimensional reaction diffusion equation with source termu_t = A₁(u)u_xx + A₂(u)u_yy + A₃(u)u_zz + B₁(u)u_x² + B₂(u)u_y² + B₃(u)u_z² + Q(u)by constructing the functional invariant setE₀ = {u : u_x = f(t)w_xF(u),u_y = f(t)w_yF(u),u_z = f(t)w_zF(u)}. Secondly, we also discuss (2+l)-dimensional wave equationu_tt= A(u)u_xx + B(u)u_yy + C(u)u_x² + D(u)u_y² + Q(u), and their solutions in terms of the invariant setS₂ = {u : u_x = v_xF(u), u_y = v_yF(u)},and we get some exact solutions of them, where w is a smooth function of vari-ables x, y, z, and v is a smooth function of variables x, y, F(u) is a smooth func-tion to be determined, A₁(u), A₂(u), A₃(u), B₁(u), B₂(u), B₃(u), A(u), B(u), C(u), D(u), Q(u) are the smooth functions of u. We also use E₀ to deal with (N+1)-dimensional reaction diffusion equation with source term. At the same time, using S₂, we simply discuss (N + 1)-dimensional wave equation.