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Register a new userLifespan of semilinear generalized Tricomi equation with Strauss type exponent
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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 to the generalized Tricomi equation.
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In this paper, we study the blow-up of solutions for semilinear wave equations with scale-invariant dissipation and mass in the case in which the model is somehow wave-like. A Strauss type critical exponent is determined as the upper bound for the exponent in the nonlinearity in the main theorems. Two blow-up results are obtained for the sub-critical case and for the critical case, respectively. In both cases, an upper bound lifespan estimate is given.
It is believed or conjectured that the semilinear wave equations with scattering space dependent damping admit the Strauss critical exponent, see Ikehata-Todorova-Yordanov cite{ITY}(the bottom in page 2) and Nishihara-Sobajima-Wakasugi cite{N2}(conjecture iii in page 4). In this work, we are devoted to showing the conjecture is true at least when the decay rate of the space dependent variable coefficients before the damping is larger than 2. Also, if the nonlinear term depends only on the derivative of the solution, we may prove the upper bound of the lifespan is the same as that of the solution of the corresponding problem without damping. This shows in another way the lqlq hyperbolicity of the equation.
The blow up problem of the semilinear scale-invariant damping wave equation with critical Strauss type exponent is investigated. The life span is shown to be: $T(varepsilon)leq Cexp(varepsilon^{-2p(p-1)})$ when $p=p_S(n+mu)$ for $0<mu<frac{n^2+n+2}{n+2}$. This result completes our previous study cite{Tu-Lin} on the sub-Strauss type exponent $p<p_S(n+mu)$. Our novelty is to construct the suitable test function from the modified Bessel function. This approach might be also applied to the other type damping wave equations.
In this paper, we are concerned with the global Cauchy problem for the semilinear generalized Tricomi equation $partial_t^2 u-t^m Delta u=|u|^p$ with initial data $(u(0,cdot), partial_t u(0,cdot))= (u_0, u_1)$, where $tgeq 0$, $xin{mathbb R}^n$ ($nge 3$), $minmathbb N$, $p>1$, and $u_iin C_0^{infty}({mathbb R}^n)$ ($i=0,1$). 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 $p_{text{conf}}(m,n)> p_{text{crit}}(m,n)$ such that the solution $u$ exists globally when $p>p_{text{conf}}(m,n)$ provided that the initial data is small enough. In case $p_{text{crit}}(m,n)<pleq p_{text{conf}}(m,n)$, we will establish global existence of small data solutions $u$ in a subsequent paper.
We concern the blow up problem to the scale invariant damping wave equations with sub-Strauss exponent. This problem has been studied by Lai, Takamura and Wakasa (cite{Lai17}) and Ikeda and Sobajima cite{Ikedapre} recently. In present paper, we extend the blowup exponent from $p_F(n)leq p<p_S(n+2mu)$ to $1<p<p_S(n+mu)$ without small restriction on $mu$. Moreover, the upper bound of lifespan is derived with uniform estimate $T(varepsilon)leq Cvarepsilon^{-2p(p-1)/gamma(p,n+2mu)}$. This result extends the blowup result of semilinear wave equation and shows the wave-like behavior of scale invariant damping wave equations solution even with large $mu>1$.
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