On the existence and cusp singularity of solutions to semilinear generalized Tricomi equations with discontinuous initial data

@article{Ruan2012OnTE,
  title={On the existence and cusp singularity of solutions to semilinear generalized Tricomi equations with discontinuous initial data},
  author={Zhuoping Ruan and Ingo Witt and Huicheng Yin},
  journal={Communications in Contemporary Mathematics},
  year={2012},
  volume={17},
  pages={1450028}
}
In this paper, we are concerned with the local existence and singularity structures of low regularity solution to the semilinear generalized Tricomi equation with typical discontinuous initial data (u(0, x), ∂tu(0, x)) = (0, φ(x)), where m ∈ ℕ, x = (x1,…,xn), n ≥ 2, and f(t, x, u) is C∞ smooth on its arguments. When the initial data φ(x) is homogeneous of degree zero or piecewise smooth along the hyperplane {t = x1 = 0}, it is shown that the local solution u(t, x) ∈ L∞([0, T] × ℝn) exists and… 
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References

SHOWING 1-10 OF 21 REFERENCES
The Cauchy problem for semilinear hyperbolic systems with discontinuous data
On demontre l'existence a t court des solutions pour le probleme de Cauchy strictement hyperbolique: Lu≡∂ t u+Σ l=1 n A l (t,x)∂ xt u=F(t,x,u(t,x)); u| t=0 =g ou g est lisse par morceaux avec des
Mixed Type Partial Differential Equations with Initial and Boundary Values in Fluid Mechanics
This paper includes various parts of the theory of mixed type partial differential equations with initial and boundary conditions in fluid mechanics ,such as: The classical dynamical equation of
On closed boundary value problems for equations of mixed elliptic‐hyperbolic type
For partial differential equations of mixed elliptic‐hyperbolic type we prove results on existence and existence with uniqueness of weak solutions for closed boundary value problems of Dirichlet and
Global Existence for the n-Dimensional Semilinear Tricomi-Type Equations
In this article we investigate the issue of global existence of the solutions of the Cauchy problem for semilinear Tricomi-type equations in ℝ n+1, n > 1. We give some sufficient conditions for
ENERGY ESTIMATES FOR WEAKLY HYPERBOLIC SYSTEMS OF THE FIRST ORDER
For a class of first-order weakly hyperbolic pseudo-differential systems with finite time degeneracy, well-posedness of the Cauchy problem is proved in an adapted scale of Sobolev spaces. These
MIXED EQUATIONS AND TRANSONIC FLOW
This paper reviews the present situation with existence and uniqueness theorems for mixed equations and their application to the problems of transonic flow. Some new problems are introduced and
On the Cauchy problem of degenerate hyperbolic equations
In this paper, we study a class of degenerate hyperbolic equations and prove the existence of smooth solutions for Cauchy problems. The existence result is based on a priori estimates of Sobolev
...
1
2
3
...