On Initial-boundary-value Problems for a Class of Systems of Quasi-linear Evolution Equations (Journal of Partial Differential Equations, 2020)
In this paper the initial-boundary-value problems for pseudo-hyperbolic system of quasi-linear equations: {(-1)^Mu_{tt} + A(x, t, U, V)u_x^{2M}_{tt} = B(x, t, U, V)u_x^{2M}_{t} + C(x, t, U, V)u_x^{2M} + f(x, t, U, V) u_x^k(0,t) = &#968_{0k}(t), \quad u_x^k(l,t) = &#968_{lk}(t), \quad k = 0,1,&#8230,M - 1 -u(x,0) = &#966_0(x), \quad u_t(x,0) = &#966_1(x) is studied, where U = (u_1, u_x,&#8230,u_x^{2M - 1}) V = (u_t, u_{xt},&#8230,u_x^{2M - 1_t}), A, B, C are m &#215 m matrices, u, f, &#968_{0k}, &#968_{1k}, &#968_0, &#968_1 are m-dimensional vector functions. The existence and uniqueness of the generalized solution (in H&sup2 (0, T; H^{2M} (0, 1))) of the problems are proved.
Keywords: Pseudo-hyperbolic system of quasi-linear equations of higher order