Weakly hyperbolic equations, Sobolev spaces of variable order, and propagation of singularities (Q696163)

If we compare the propagation of singularities for solutions of Cauchy problems for \(\square u=f(u)\), \(\square v=0\) with the same data \(\varphi \in H^s\), \(\psi\in H^{s-1}\), then \(u-v\in C([0,T], H^{s+1})\). Consequently, singularities of \(u\) formed by influence of the semilinear term are weaker than those of \(v\) at least by one Sobolev order. The goal of the present paper is to show a corresponding result for \(Lu=f(t,x,u)\), \(Lv=0\) with the same Cauchy data, where \[ L=D^2_t+2 \sum^n_{j=1} t^{l_*}c_j (t,x)D_tD_{x_j}-\sum^n_{i,j=1} t^{2l_*} a_{ij} (t ,x)D_{x_i}D_{x_j} \] \[ -\sum^n_{j=1} il_*t^{l_*-1}b_j(t,x) D_{x_j}+ ic_0 (t,x)D_t,\;l_*\in\mathbb{N}, \] is a weakly hyperbolic operator with smooth coefficients, where the \(C^\infty\)-type Levi condition is satisfied. The main difficulty the author has to overcome is the so-called loss of derivatives, that is, the Sobolev regularity of the solution is less than those of the data. This loss depends among other things on the coefficients. Consequently, there appears a variable loss of regularity depending on spatial variables. For this reason the author introduces weighted Sobolev spaces of variable order, where the weight contains the variable loss of derivatives. These Sobolev spaces form algebras. This property is used to include at least (analytic) nonlinearities in the model. It would be interesting to include nonanalytic nonlinearities, too. But then composition operators should be studied in the above-mentioned Sobolev spaces. The reason why the author can understand propagation of strongest singularities is that the very tricky choice of function spaces allows to prove a priori estimates of strictly hyperbolic type for \(Lu=f(t,x)\) after application of Duhamel's principle and construction of the fundamental solution. The construction of the fundamental solution (parametrix) follows the usual concept for weakly hyperbolic operators. It would be as well to know if the results can be transferred to the weakly hyperbolic case with infinite degeneracy.