This paper investigates a weakly coupled system of semilinear Euler-Poisson-Darboux-Tricomi equations (EPDTS) with power-type nonlinear terms. More precisely, in the case where the damping terms dominate over the mass terms, the critical curve in the $p-q$ plane that delineates the threshold between global existence and blow-up for the EPDTS is given by \begin{equation*} Γ_m(n,p,q,β_1,β_2)=0, \end{equation*} where $Γ_m$ is defined by (\ref{gammam}). Through the construction of new test functions, the blow-up problem is addressed when $Γ_m(n,p,q,β_1,β_2)\geq0$. Based on the $(L^1\cap L^2)-L^2$ estimates of the solution to the corresponding linear equation established in our previous work \cite{LiGuo2025}, we derive the global existence of solutions with small initial data when $Γ_m(n,p,q,β_1,β_2)<0$, provided that the damping terms prevail over the mass terms.
In this paper, we consider the blow-up problem of semilinear generalized Tricomi equation. Two blow-up results with lifespan upper bound are obtained under subcritical and critical Strauss type exponent. In the subcritical case, the proof is based on the test function method and the iteration argument. In the critical case, an iteration procedure with the slicing method is employed. This approach has been successfully applied to the critical case of semilinear wave equation with perturbed Laplacian or the damped wave equation of scattering damping case. The present work gives its application t…
We prove the local existence and uniqueness of minimal regularity solutions $u$ of the semilinear generalized Tricomi equation $\partial_t^2 u-t^m Δu =F(u)$ with initial data $(u(0,\cdot), \partial_t u(0,\cdot)) \in \dot{H^γ}(\mathbb R^n) \times \dot{H}^{γ-\frac2{m+2}}(\mathbb R^n)$ under the assumption that $|F(u)|\lesssim |u|^κ$ and $|F'(u)| \lesssim |u|^{κ-1}$ for some $κ>1$. Our results improve previous results of M. Beals [2] and of ourselves [15-17]. We establish Strichartz-type estimates for the linear generalized Tricomi operator $\partial_t^2 -t^m Δ$ from which the semilinear results …
In [19-20], we have established the existence and singularity structures of low regularity solutions to the semilinear generalized Tricomi equations in the degenerate hyperbolic regions and to the higher order degenerate hyperbolic equations, respectively. In the present paper, we shall be concerned with the low regularity solution problem for the semilinear mixed type equation $\p_t^2u-t^{2l-1}Δu= f(t,x,u)$ with an initial data $u(0,x)=φ(x)\in H^{s}(\Bbb R^n)$ ($0\le s<\f{n}{2}$), where $(t,x)\in\Bbb R \times\Bbb R^n$, $n\ge 2$, $l\in\Bbb N$, $f(t,x,u)$ is $C^1$ smooth in its arguments and ha…
In this paper, we are concerned with the global Cauchy problem for the semilinear generalized Tricomi equation $\partial_t^2 u-t^m Δu=|u|^p$ with initial data $(u(0,\cdot), \partial_t u(0,\cdot))= (u_0, u_1)$, where $t\geq 0$, $x\in{\mathbb R}^n$ ($n\ge 3$), $m\in\mathbb N$, $p>1$, and $u_i\in C_0^{\infty}({\mathbb R}^n)$ ($i=0,1$). We show that there exists a critical exponent $p_{\text{crit}}(m,n)>1$ such that the solution $u$, in general, blows up in finite time when $1<p<p_{\text{crit}}(m,n)$. We further show that there exists a conformal exponent $p_{\text{conf}}(m,n)> p_{\text{crit}}(m,n…
In the present paper, we investigate the blow-up dynamics for local solutions to the semilinear generalized Tricomi equation with combined nonlinearity. As a result, we enlarge the blow-up region in comparison to the ones for the corresponding semilinear models with either power nonlinearity or nonlinearity of derivative type. Our approach is based on an iteration argument to establish lower bound estimates for the space average of local solutions. Finally, we obtain upper bound estimates for the lifespan of local solutions as byproduct of our iteration argument.
The Tricomi equation is a second-order partial differential equation of mixed elliptic-hyperbolic type. It was first analyzed in the work by Francesco Giacomo Tricomi (1923) on the well-posedness of a boundary value problem. The Tricomi equation can be transformed into the corresponding elliptic or hyperbolic Euler-Poisson-Darboux equation, and has a close connection with transonic flow and isometric embedding. It has different degeneracy from a closely related equation, the Keldysh equation.
We present some integral transform that allows to obtain solutions of the generalized Tricomi equation from solutions of a simpler equation. We used in [13,14],[41]-[46] the particular version of this transform in order to investigate in a unified way several equations such as the linear and semilinear Tricomi equations, Gellerstedt equation, the wave equation in Einstein-de Sitter spacetime, the wave and the Klein-Gordon equations in the de Sitter and anti-de Sitter spacetimes.
For 1-D semilinear Tricomi equation $\partial_t^2 u-t\partial_x^2u=|u|^p$ with initial data $(u(0,x), \partial_t u(0,x))$ $=(u_0(x), u_1(x))$, where $t\ge 0$, $x\in\mathbb{R}$, $p>1$, and $u_i\in C_0^\infty(\mathbb{R})$ ($i=0,1$), we shall prove that there exists a critical exponent $p_{\rm crit}=5$ such that the small data weak solution $u$ exists globally when $p>p_{\rm crit}$; on the other hand, the weak solution $u$, in general, blows up in finite time when $1<p<p_{\rm crit}$. We specially point out that for 1-D semilinear wave equation $\partial_t^2 v-\partial_x^2v=|v|^p$, the weak soluti…
We study in this paper the small data Cauchy problem for the semilinear generalized Tricomi equations with a nonlinear term of derivative type $u_{tt}-t^{2m}Δu=|u_t|^p$ for $m\ge0$. Blow-up result and lifespan estimate from above are established for $1<p\le 1+\frac{2}{(m+1)(n-1)-m}$. If $m=0$, our results coincide with those of the semilinear wave equation. The novelty consists in the construction of a new test function, by combining cut-off functions, the modified Bessel function and a harmonic function. Interestingly, if $n=2$ the blow-up power is independent of $m$. We also furnish a local …
In this note, we prove a blow-up result for a semilinear generalized Tricomi equation with nonlinear term of derivative type, i.e., for the equation $\mathscr{T}_{\!\!\ell} u = |\partial_t u|^p$, where $ \mathscr{T}_{\!\!\ell} = \partial_t^2-t^{2\ell}Δ$. Smooth solutions blow up in finite time for positive Cauchy data when the exponent $p$ of the nonlinear term is below $\frac{\mathscr{Q}}{\mathscr{Q}-2}$, where $\mathscr{Q}=(\ell+1)n+1$ is the quasi-homogeneous dimension of the generalized Tricomi operator $\mathscr{T}_{\!\!\ell}$. Furthermore, we get also an upper bound estimate for the life…
In this paper, we are concerned with the local existence and singularity structure of low regularity solutions to the semilinear generalized Tricomi equation $\p_t^2u-t^mΔu=f(t,x,u)$ with typical discontinuous initial data $(u(0,x), \p_tu(0,x))=(0, \vp(x))$; here $m\in\Bbb N$, $x=(x_1, ..., x_n)$, $n\ge 2$, and $f(t,x,u)$ is $C^{\infty}$ smooth in its arguments. When the initial data $\vp(x)$ is a homogeneous function of degree zero or a piecewise smooth function singular along the hyperplane ${t=x_1=0}$, it is shown that the local solution $u(t,x)\in L^{\infty}([0,T]\times\Bbb R^n)$ exists an…
