The self-similar solutions of the Tricomi-type equations

Abstract

This paper analyses the properties of the family of self-similar solutions of the generalized Tricomi equation utt − t 2k △ u = 0 (2k ∈ N) in the domain R 1+n + by considering initial conditions on the functions and their derivatives, posed as the Cauchy problem with homogeneous initial data. For specific values of the power k (= 1/2 or = 3/2) and n = 1 this problem has applications in the aerodynamics of airfoils operating in transonic flows of perfect or dense gases, respectively. An integral transformation is suggested and used to represent the solutions of the Cauchy problem with homogeneous initial functions in terms of fundamental solutions of the classical wave equation (the case k = 0). Then the Cauchy problem with homogeneous initial functions for the wave equation in R 1+n is solved. These results are used to derive estimates of the upper bound for solutions' size and to obtain the asymptotics for self-similar solutions of the wave equation and of the Tricomi-type equation in the neighbourhood of their light cones.

Self-similar solutions of semilinear wave equation with variable speed of propagation

“…[29].) Consider the Cauchy problem for the Tricomi-type equation (1.7), (2.1) with the homogeneous data…”

“…The order of singularity on the light cone depends on a and b, and is given by the next theorem. [29].) Consider the Cauchy problem for the Tricomi-type equation (1.7), (2.1) with the homogeneous data ϕ 0 ∈ C ∞ (R n * ) and ϕ 1 ∈ C ∞ (R n * ) of order −a and −b, respectively.…”

“…In Section 2 we quote from [29] results on the linear Tricomi-type equation, which describe the order of singularity on the light cone and the decay at infinity, |x| → ∞, of the self-similar solutions of the Cauchy problem with homogeneous initial functions. In Section 3 we prove the decay (in t → ∞) estimates in the Sobolev and Besov spaces for the solutions of linear Tricomi-type equation without source.…”

On the strauss index of semilinear tricomi equation

“…t u − t m ∆u = |u| p , (t, x) ∈ R 1+n + , u(0, x) = u 0 (x), ∂ t u(0, x) = u 1 (x), (1.3) where m ∈ N, p > 1, n ≥ 2. For the local existence and regularity of solution u to (1.3) under weak regularity assumptions on (u 0 , u 1 ), the reader may consult [24,25,26,34,35]. If u i ∈ C ∞ 0 (R n ) (i = 0, 1) Yagdjian in [34] proved the blowup and global existence of u when p belongs to different intervals.…”

“…There are extensive results for both linear and semilinear Tricomi equations in n space dimensions (n ∈ N). For instances, the authors of [1,32,35] have computed the forward fundamental solution of the linear Tricomi equation ∂ 2 t u − t∆u = 0 explicitly. The authors of [11,19,20,21,22] have obtained a series of interesting results on the existence and uniqueness of solutions u to the semilinear Tricomi equation ∂ 2 t u − t∆u = f (t, x, u) in bounded domains, under certain restrictions on the nonlinearity f (t, x, u).…”