[Paper Review] On the global solution problem for semilinear generalized Tricomi equations, I

Daoyin He, Ingo Witt+1 more|arXiv (Cornell University)|Nov 27, 2015

Advanced Mathematical Physics Problems23 references3 citations

TL;DR

This paper establishes the existence of a critical exponent $ p_{\text{crit}}(m,n) > 1 $ for the semilinear generalized Tricomi equation $ \partial_t^2 u - t^m \Delta u = |u|^p $, showing finite-time blowup for $ 1 < p < p_{\text{crit}}(m,n) $, and a conformal exponent $ p_{\text{conf}}(m,n) > p_{\text{crit}}(m,n) $ such that small data solutions exist globally when $ p > p_{\text{conf}}(m,n) $, using Strichartz estimates and iterative $ L^p $-based arguments in a weighted function space framework.

ABSTRACT

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 $t\geq 0$, $x\in{\mathbb R}^n$ ($n\ge 3$), $m\in\mathbb N$, $p&gt;1$, and $u_i\in C_0^{\infty}({\mathbb R}^n)$ ($i=0,1$). We show that there exists a critical exponent $p_{ ext{crit}}(m,n)&gt;1$ such that the solution $u$, in general, blows up in finite time when $1 p_{ ext{crit}}(m,n)$ such that the solution $u$ exists globally when $p&gt;p_{ ext{conf}}(m,n)$ provided that the initial data is small enough. In case $p_{ ext{crit}}(m,n)

Motivation & Objective

• To determine a critical exponent $ p_{\text{crit}}(m,n) > 1 $ such that solutions to the semilinear generalized Tricomi equation blow up in finite time when $ 1 < p < p_{\text{crit}}(m,n) $.
• To identify a conformal exponent $ p_{\text{conf}}(m,n) > p_{\text{crit}}(m,n) $ such that small data solutions exist globally in time when $ p > p_{\text{conf}}(m,n) $.
• To lay the analytical foundation for proving global existence of small data solutions in the intermediate range $ p_{\text{crit}}(m,n) < p \leq p_{\text{conf}}(m,n) $, which is addressed in a subsequent paper.
• To extend the Strauss conjecture framework to the time-dependent coefficient case $ t^m \Delta u $, generalizing results from the standard wave and dissipative wave equations.

Proposed method

• The authors employ a sequence of iterative approximations $ u_k $ to construct a solution in $ L^r $-based function spaces with weights depending on $ m $ and $ n $.
• They use a weighted $ L^p $-norm involving the Riesz potential $ |D_x|^{\sigma} $ to control the regularity of nonlinear terms.
• A key step involves applying a variant of the Kato-type inequality (4.12) to estimate the nonlinearity $ |u_k|^p $ in terms of fractional derivatives of $ u_k $.
• The proof relies on uniform boundedness of the sequence $ M_k $, defined as a supremum of weighted $ L^q $-norms of $ u_k $, under small data assumptions.
• Convergence of the sequence $ u_k $ in $ L^{q_0} $ is established via a contraction argument using the bound $ N_{k+1} \leq \frac{1}{2} N_k $, ensuring strong convergence in the limit.
• The analysis combines Strichartz-type estimates adapted to the generalized Tricomi operator and interpolation techniques in Lorentz-type spaces.

Experimental results

Research questions

• RQ1 What is the critical exponent $ p_{\text{crit}}(m,n) $ that separates finite-time blowup from potential global existence for the semilinear generalized Tricomi equation?
• RQ2 Does there exist a conformal exponent $ p_{\text{conf}}(m,n) > p_{\text{crit}}(m,n) $ such that small data solutions exist globally when $ p > p_{\text{conf}}(m,n) $?
• RQ3 How does the time-dependent coefficient $ t^m $ in the principal part affect the existence and blowup behavior compared to the standard wave equation?
• RQ4 Can the Strauss conjecture framework be extended to equations with variable coefficients of the form $ t^m \Delta u $?
• RQ5 What is the precise threshold for global existence in the intermediate range $ p_{\text{crit}}(m,n) < p \leq p_{\text{conf}}(m,n) $, and how is it approached via iterative methods?

Key findings

• A critical exponent $ p_{\text{crit}}(m,n) > 1 $ exists such that solutions to the generalized Tricomi equation typically blow up in finite time when $ 1 < p < p_{\text{crit}}(m,n) $.
• A conformal exponent $ p_{\text{conf}}(m,n) > p_{\text{crit}}(m,n) $ exists such that small data solutions exist globally in time when $ p > p_{\text{conf}}(m,n) $.
• The critical exponent satisfies $ p_{\text{crit}}(m,n) = \frac{(m+2)(n-2)+6}{(m+2)(n-2)-2} $ when $ p $ is an integer and $ p \geq \frac{(m+2)(n-2)+6}{(m+2)(n-2)-2} $.
• For $ p > p_{\text{conf}}(m,n) $, the solution $ u $ is shown to exist globally via an iterative scheme converging in $ L^{q_0} $, with uniform bounds on the sequence $ M_k $.
• The proof relies on a refined Strichartz-type estimate and a fractional derivative inequality (4.12) to control the nonlinearity $ |u|^p $ in the iterative construction.
• The convergence of the iterative sequence $ u_k $ to a solution $ u $ is established via a contraction argument, yielding $ \|u_{k+1} - u_k\|_{L^{q_0}} \leq \frac{1}{2} \|u_k - u_{k-1}\|_{L^{q_0}} $ for small data.