Levi condition and analytic regularity for quasilinear weakly hyperbolic equations of second order
In the theory of nonlinear weakly hyperbolic equations an interesting question is that for the global regularity of solutions. Let us consider the strictly hyperbolic Cauchy problem \((a(x, t)\geq \varepsilon> 0)\), \[ u_{tt}- (a(t, x) u_x)_x= f(t, x, u, u_t, u_x),\;u(x, 0)= u_0(x),\;u_t(x, 0)= u_1(x).\tag{1} \] Under assumptions of analyticity the analytic regularity can be proved, that is, if the solution belongs to some Sobolev space of minimal regularity, then it is automatically analytic. If we weaken the assumption for \(a= a(x, t)\) to \(a(x, t)\geq 0\) (weakly hyperbolic case), then at least two new difficulties appear: 1. One has to take account for the set of zeros of the leading coefficient \(a(x, t)\). For this reason differential inequalities for \(a(t, x)\) are prescribed in various papers. 2. One has to formulate Levi conditions, these are conditions between derivatives of \(f\) with respect to \(u_x\) and \(a(t, x)\). In the present paper, the authors consider (1) in the weakly hyperbolic case. Instead of differential inequalities for \(a(t, x)\) the analyticity condition guarantees a good behaviour of the set of zeros. By the Weierstrass preparation theorem the so-called cusp condition is satisfied. Together with a nonlinear Levi condition energy estimates on cusp shaped domains are proved. The property of existence of cone of dependence for the solutions and the Cauchy-Kovalevsky theorem lead to an analytic regularity result. The case of Gevrey regularity (Gevrey order is smaller than the critical order 2) is discussed, too.
scientific article; zbMATH DE number 808304