Critical exponent for damped wave equations with nonlinear memory (1004.3850v4)

Abstract: We consider the Cauchy problem in $\mathbb{R}^n,$ $n\geq 1,$ for a semilinear damped wave equation with nonlinear memory. Global existence and asymptotic behavior as $t\rightarrow\infty$ of small data solutions have been established in the case when $1\leq n\leq3.$ Moreover, we derive a blow-up result under some positive data in any dimensional space.

• The paper establishes the critical exponent for a semilinear damped wave equation with nonlinear memory by analyzing local well-posedness and conditions for finite time blow-up.
• It uses weighted energy methods to prove global existence for small, compactly supported data in low spatial dimensions under specific parameter ranges.
• The study reveals that the fractional memory term critically influences energy decay rates and solution behavior, offering valuable insights for numerical simulation and physical modeling.

This paper (1004.3850) investigates the Cauchy problem for a semilinear damped wave equation with a nonlinear memory term in $\mathbb{R}^n$, $n \ge 1$: $$u_{tt} - \Delta u + u_t = \int_0^t (t-s)^{-\gamma} |u(s)|^p ds$$, for $t>0, x \in \mathbb{R}^n$, with initial conditions $u(0,x) = u_0(x)$ and $u_t(0,x) = u_1(x)$. The parameters are $0 < \gamma < 1$ and $p > 1$. The nonlinear memory term is a Riemann-Liouville fractional integral of $|u|^p$, specifically $J_0^{1-\gamma}(|u|^p)(t)$. The paper focuses on understanding the conditions on $p$ and $\gamma$ that determine whether solutions exist globally or blow up in finite time, particularly identifying a "critical exponent" analogous to the Fujita exponent for reaction-diffusion equations or critical exponents for standard semilinear damped wave equations.

The initial data are assumed to be in $H^1(\mathbb{R}^n) \times L^2(\mathbb{R}^n)$ and compactly supported within a ball $B(K)$.

The paper establishes the following key results:

1. Local Well-posedness (Proposition 1): For suitable ranges of $p$ and $\gamma$, the problem has a unique maximal mild solution in $$C([0, T_{max}), H^1(\mathbb{R}^n)) \cap C^1([0, T_{max}), L^2(\mathbb{R}^n))$$. Solutions maintain the finite propagation speed property, meaning their support remains within $B(t+K)$. Blow-up in finite time ($T_{max} < \infty$) occurs if and only if the $H^1 \times L^2$ norm of the solution tends to infinity as $t \to T_{max}$.
2. Global Existence (Theorem 1): For spatial dimensions $1 \le n \le 3$, specific ranges of $\gamma$ ($(1/2, 1)$ for $n=1,2$ and $(11/16, 1)$ for $n=3$), and sufficiently small initial data (in $H^1 \times L^2$ norm, compactly supported), if the exponent $p$ is larger than certain dimension-dependent critical values ($p_1, p_2, p_3$ for $n=1, 2, 3$ respectively), then the solution exists globally in time ($T_{max} = \infty$).
   • $p_1 = 1 + \frac{2(3-2\gamma)}{n-2+2\gamma}$ (for $n=1$ with specific $\gamma$ range)
   • $p_2 = 1 + \frac{4(3-2\gamma)}{n-4+4\gamma}$ (for $n=2$ with specific $\gamma$ range)
   • $p_3 = 1 + \frac{2(5-4\gamma)}{n-2+4\gamma}$ (for $n=3$ with specific $\gamma$ range) The proof relies on a weighted energy method, showing that a suitably defined weighted energy functional remains bounded for small initial data.
3. Finite Time Blow-up (Theorem 2): Under the assumption that the initial data $(u_0, u_1)$ have positive integrals over $\mathbb{R}^n$ ($\int u_i(x) dx > 0$ for $i=0,1$), blow-up in finite time occurs if $p$ is less than or equal to a value $p_\gamma = 1 + \frac{2(2-\gamma)}{n-2+2\gamma}$, provided $(n-2)/n < \gamma < 1$. For $n \ge 3$ and $\gamma \le (n-2)/n$, blow-up also occurs if $p \le n/(n-2)$. The authors conjecture that $p_\gamma$ (or $p_1$ in the $n=1$ case) acts as the critical exponent for the equation, drawing parallels to heat equations with memory. The proof uses the test function method.
4. Asymptotic Behavior (Theorem 3): For the global small data solutions obtained under the conditions of Theorem 1, the paper provides estimates for the large time behavior. Solutions decay exponentially outside any ball $B(t^{1/2+\delta})$ for $\delta > 0$. The total energy ($|Du(t,.)|_2^2$) decays polynomially, with rates depending on $n$ and $\gamma$:
   • $O(t^{-(n/4+1/2-\gamma)})$ for $n=1$.
   • $O(t^{-(1/2-\gamma)})$ for $n=2$.
   • $O(t^{-\gamma})$ for $n=3$. These rates show how the memory term affects the energy decay compared to the standard damped wave equation.

Implementation and Application Notes:

• Modeling: The equation is relevant for modeling physical systems exhibiting both damping and memory effects, where the rate of change and spatial diffusion are influenced not just by the current state but by the history of the solution. This could include certain types of viscoelastic materials, heat conduction in materials with memory, or population dynamics with delayed effects.
• Numerical Simulation: Simulating this equation numerically would require handling the fractional integral term and the wave dynamics with damping.
  • Discretization: A finite difference or finite element method can be used for spatial discretization.
  • Time Stepping: Due to the wave term, explicit methods might be subject to Courant-Friedrichs-Lewy (CFL) stability conditions, potentially requiring small time steps. The damping term can sometimes relax CFL constraints but may introduce stiffness. Implicit methods could be more stable but computationally more expensive per step.
  • Memory Term: The memory integral $\int_0^t (t-s)^{-\gamma} |u(s)|^p ds$ requires storing the history of $|u(s)|^p$ from $s=0$ to the current time $t$. This can be computationally expensive and memory-intensive for long time simulations. Numerical methods for fractional integrals (e.g., product integration rules) would be needed. The singularity at $s=t$ must be handled carefully.
  • Nonlinearity: The term $|u(s)|^p$ introduces nonlinearity, potentially requiring iterative solvers (e.g., Newton's method) within each time step if using implicit schemes.
  • Blow-up: If simulating scenarios where blow-up is predicted (based on Theorem 2), numerical solutions will likely develop sharp gradients and rapidly increasing norms, eventually becoming intractable or exhibiting numerical instability if not handled with adaptive methods.
• Critical Exponent Significance: The critical exponent $p_\gamma$ (and $p_n$) is a crucial design parameter for physical systems. If $p$ is below the critical value, solutions tend to blow up, indicating potential instability. If $p$ is above the critical value (and initial data are small), solutions are globally stable and decay. This knowledge can guide material design or system parameter choices to ensure desired stability or behavior.
• Practical Implementation Challenges:
  • Memory Cost: Storing the full history for the fractional integral can be prohibitive. Techniques like approximating the kernel or using specialized numerical schemes for fractional differential equations that don't require full history storage might be necessary.
  • Computational Cost: Solving systems involving fractional operators is generally more expensive than integer-order counterparts.
  • Parameter Tuning: The results depend on $n, \gamma, p$. Practical applications would involve estimating these parameters from experimental data.

In summary, this paper provides rigorous mathematical results on the well-posedness, global existence, blow-up, and asymptotic behavior of a damped wave equation with nonlinear memory, identifying critical exponents that govern these behaviors. Translating these findings into practical applications involves careful numerical treatment of the fractional and nonlinear terms, considering the computational and memory costs associated with the memory term, and using the critical exponent results to inform system design or analysis.