[Paper Review] Blow-up and lifespan estimates for Nakao's type problem with nonlinearities of derivative type

Wenhui Chen|arXiv (Cornell University)|May 4, 2020

Advanced Mathematical Physics Problems22 references5 citations

TL;DR

This paper establishes blow-up and lifespan estimates for a semilinear hyperbolic system of Nakao's type with derivative-type nonlinearities, using an iteration method with time-dependent functionals and slicing. The key result shows the model remains hyperbolic-like, with lifespan upper bound $ T(\varepsilon) \lesssim \varepsilon^{-1/\max(T_1,T_2)} $, where $ T_1, T_2 $ depend on space dimension $ n $ and nonlinear exponents $ p,q $, and the blow-up condition is optimal in one dimension.

ABSTRACT

In the present paper, we investigate blow-up and lifespan estimates for a class of semilinear hyperbolic coupled system in $\mathbb{R}^n$ with $n\geqslant 1$, which is part of the so-called Nakao's type problem weakly coupled a semilinear damped wave equation with a semilinear wave equation with nonlinearities of derivative type. By constructing two time-dependent functionals and employing an iteration method for unbounded multiplier with slicing procedure, the results of blow-up and upper bound estimates for the lifespan of energy solutions are derived. The model seems to be hyperbolic-like instead of parabolic-like. Particularly, the blow-up result for one dimensional case is optimal.

Motivation & Objective

• To investigate blow-up and lifespan of energy solutions for a weakly coupled system of damped and undamped wave equations with derivative-type nonlinearities.
• To determine whether such nonlinearities shift the model from hyperbolic-like to parabolic-like behavior, as seen in damped wave equations.
• To derive sharp upper bounds for the lifespan of local solutions under small initial data.
• To extend the iteration method with time-dependent functionals and slicing to unbounded multipliers in the context of derivative nonlinearities.
• To confirm the optimality of the blow-up condition in the one-dimensional case.

Proposed method

• Construction of two time-dependent functionals to track energy growth in the iteration process.
• Application of an iteration method for unbounded multipliers, adapted to handle time-dependent weights and slicing in time intervals.
• Use of slicing procedure to manage the growth of norms over successive time intervals, enabling iterative estimates.
• Employment of L’Hôpital’s rule and asymptotic analysis to derive lower bounds for sequences $ D_j $ and $ Q_j $, which control the functional growth.
• Derivation of recursive inequalities for $ D_j $ and $ Q_j $, leading to exponential growth in $ (pq)^{j/2} $, indicating finite-time blow-up.
• Estimation of the lifespan via blow-up of lower bounds for functionals $ F_1(t) $ and $ F_2(t) $ as $ t \to \infty $, leading to $ T(\varepsilon) \lesssim \varepsilon^{-1/\max(T_1,T_2)} $.

Experimental results

Research questions

• RQ1 Does the presence of time-derivative nonlinearities in both equations shift the model from hyperbolic-like to parabolic-like behavior, as in classical damped wave equations?
• RQ2 What is the sharp lifespan estimate for energy solutions of Nakao’s type problem with derivative-type nonlinearities?
• RQ3 How does the critical exponent for blow-up compare to the Glassey exponent in the absence of damping?
• RQ4 Is the blow-up condition optimal in one space dimension?
• RQ5 Can the iteration method with slicing and time-dependent functionals be extended to unbounded multipliers in systems with derivative nonlinearities?

Key findings

• The blow-up condition for Nakao’s type problem with derivative-type nonlinearities is governed by the Glassey exponent, confirming hyperbolic-like behavior.
• The lifespan of energy solutions satisfies $ T(\varepsilon) \lesssim \varepsilon^{-1/\max(T_1(p,q,n), T_2(p,q,n))} $, with $ T_1, T_2 $ explicitly defined in terms of $ n, p, q $.
• In one dimension, the blow-up condition is optimal, matching the known sharp threshold for the Glassey exponent.
• The model remains hyperbolic-like even with derivative nonlinearities, as the dominant term in the iteration is driven by the wave equation component.
• The iteration method with slicing and time-dependent functionals successfully yields sharp lifespan estimates despite the unbounded nature of the multiplier.
• For $ pq < (n+1)/(n-1) $, the power of $ t $ in the exponential lower bound is positive, ensuring blow-up of the functional as $ t \to \infty $.