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Propagation of mild singularities for semilinear weakly hyperbolic equations. (English) Zbl 0964.35099

The authors prove the local existence, uniqueness and propagation of singularities of solutions to the Cauchy problem for the equations \(u_{tt}+\sum _{j=1}^nc_j(t)\lambda (t)u_{x_jt}- \sum _{i,j=1}^na_{ij}(t)\lambda (t)^2u_{x_ix_j}+ \sum _{j=1}^nb_j(t)\lambda '(t)u_{x_j}+c_0(t)u_t=\sum _{j=1}^{\infty }f_ju^j.\) Using an appropriate scale of generalized Sobolev-like spaces they introduce the framework of optimal spaces assigned to weakly hyperbolic operators. Besides, the results tell us that mild singularities of the solutions to semilinear equations propagate in the same way as that to linear equations.

MSC:

35L67 Shocks and singularities for hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations
35B45 A priori estimates in context of PDEs
35C10 Series solutions to PDEs
35A20 Analyticity in context of PDEs
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