We discuss the local existance and uniqueness of solutions of certain nonstrictly hyperbolic systems, with Hölder continuous coefficients with respect to time variable. We reduce the nonstrictly hyperbolic systems to the parabolic ones, then we shall prove them by use of Tanabe-Sobolevski's method.
We discuss the local existence and uniqueness of solutions of certain nonstrictly hyperbolic systems, with Hölder continuous coefficients with respect to time variable. We reduce the nonstrictly hyperbolic systems to the parabolic ones and by use of the Tanabe-Sobolevski's method and the Banach scale method we construct a semi-group which gives a representation of the solution to the Cauchy problem.
We consider three kind of oscillatory properties of the solutions to semilinear degenerate hyperbolic equations. Several sufficient conditions for the oscillation or non-oscillation are presented. In particular, they give us the positivity of the solutions for semilinear hyperbolic equations degenerating at initial point in one space dimension. Moreover we establish a few oscillatory conditions for the solutions of the mixed problem reduced to in one space dimension.
In this paper we shall solve locally in time the solutions to the Cauchy problem for first order quasilinear hyperbolic systems of which coefficients of principal part and of lower order terms are μ-Hölder and μ -Hölder continuous in time variable respectively and in Gevrey class of index s with respect to space variables under the assumption 1 ≤ s < min{1+ μ ν , 1+ 1−μ+μ ν }, 0 < μ ≤ 1, where ν denotes the maximal muliplicity of characteristics of systems.
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