On a system of semilinear damped $\sigma$-evolution equations with different damping types in the critical case

Technical Summary: On a System of Semilinear Damped $\sigma$-Evolution Equations with Different Damping Types in the Critical Case

Problem Statement
This paper investigates the Cauchy problem for a non-symmetric, weakly coupled system of semilinear damped $\sigma$-evolution equations in $\mathbb{R}^n$. The system is given by:
$$\begin{cases} u_{tt} + (-\Delta)^\sigma u + (-\Delta)^\sigma u_t = |v|^{p^*}\mu_1(|v|), \\ v_{tt} + (-\Delta)^\sigma v + v_t = |u|^{q^*}\mu_2(|u|), \end{cases}$$
with initial data $(u, u_t, v, v_t)(0, x) = (0, u_1, 0, v_1)(x)$. Here, $\sigma \ge 1$ and the damping terms differ: the first equation features structural damping $(-\Delta)^\sigma u_t$ (parabolic-like behavior), while the second features standard damping $v_t$ ($\sigma$-evolution-like behavior). The exponents $(p^*, q^*)$ lie on the critical curve defined by the interaction of these two distinct damping types. The nonlinearities include moduli of continuity $\mu_1$ and $\mu_2$, which provide additional regularity beyond standard power laws. The primary objective is to determine sharp conditions on $\mu_1$ and $\mu_2$ that distinguish between the global existence of small-data Sobolev solutions and finite-time blow-up.

Methodology
The authors employ two distinct analytical frameworks corresponding to the two main results:

1. Global Existence (Fixed Point Argument):

   • The proof utilizes Banach's fixed point theorem in a carefully constructed solution space $X(t)$.
   • Key to this approach is the derivation of decay estimates for the linearized problems associated with the two different damping types.
   • To handle the critical case and the non-symmetric nature of the system, the authors introduce weight functions $\ell_1(t)$ and $\ell_2(t)$ that account for the "loss of decay" in the nonlinear terms. These weights depend on the moduli of continuity and the specific position of $(p^*, q^*)$ relative to the critical thresholds.
   • Fractional Gagliardo-Nirenberg inequalities and specific integral estimates involving the moduli of continuity are used to close the contraction mapping argument.

2. Blow-up Result (Test Function Method):

   • The authors adapt the test function method to the non-symmetric, fractional setting.
   • A significant technical challenge addressed is the non-local nature of the fractional Laplacian $(-\Delta)^\sigma$ when $\sigma$ is not an integer. Standard compactly supported test functions are insufficient because the support of $(-\Delta)^\sigma \phi$ generally exceeds the support of $\phi$.
   • To overcome this, the authors construct a modified family of test functions $\phi_{j,R}$ with non-compact support, possessing specific scaling orders in space and time.
   • The integral domain is decomposed into several regions ($Q^1_{j,R}, Q^2_R, Q^3_{j,R}$) to handle the singular behavior of the auxiliary functionals near the origin and at infinity.
   • By defining auxiliary functionals $G_p$ and $G_q$ and deriving a system of ordinary differential inequalities, the authors demonstrate that the integral conditions on the moduli of continuity lead to a contradiction if a global solution were to exist.

Key Results

• Global Existence (Theorem 1.1): Under the assumption $1 < \sigma < n < 2\sigma$ and $p^* \ge 2$, the paper establishes the existence of a unique global Sobolev solution for small initial data if the moduli of continuity satisfy specific integrability conditions.

  • If the standard integrability conditions $\int_0^c \frac{\mu_j(s)}{s} ds < \infty$ hold, global existence is guaranteed.
  • If these fail, the paper provides sharp alternative conditions involving coupled integrals of $\mu_1$ and $\mu_2$ (e.g., $\int_0^c \frac{1}{s} (\mu_1(s))^{\frac{q^*}{q^*+1}} (\mu_2(s))^{\frac{1}{q^*+1}} ds < \infty$) that still ensure global existence.
  • The solution exhibits specific decay rates with a "loss of decay" factor determined by the moduli of continuity.

• Blow-up (Theorem 1.2): Conversely, if the moduli of continuity are "too weak" (i.e., the corresponding integrals diverge), the paper proves that no global Sobolev solution exists, leading to finite-time blow-up.

  • The conditions for blow-up are the divergence of the same integrals that guarantee existence in the global case.
  • This result holds for both integer and fractional $\sigma$, provided $n \ge 2\lfloor \sigma \rfloor$ in the fractional case.

• Symmetric Models (Theorem 3.2): By adapting the blow-up proof techniques, the authors resolve an open problem regarding symmetric models where both equations have the same damping type (either both structural or both standard) but with fractional parameters. They establish blow-up results for these symmetric systems in the critical case.

Significance and Contributions
The paper claims to provide a non-trivial improvement over existing literature in the following ways:

1. Non-Symmetric Critical Systems: It addresses the critical case for a system coupling "parabolic-like" and "$\sigma$-evolution-like" equations, a setting where the roles of the variables and exponents are not interchangeable, unlike in symmetric systems.
2. Fractional Laplacian Handling: The work successfully extends the test function method to systems involving fractional damping operators ($(-\Delta)^\sigma$) where $\sigma$ is fractional. This is achieved by constructing non-compactly supported test functions and carefully managing the non-local effects, a difficulty that previously limited such analyses to integer $\sigma$.
3. Sharp Thresholds: The paper determines the precise threshold for the moduli of continuity $\mu_1$ and $\mu_2$ that separates global existence from blow-up, refining the understanding of critical nonlinearity in weakly coupled systems.
4. Resolution of Open Problems: The methodology developed is successfully applied to provide a positive answer to an open problem regarding the blow-up of symmetric models with fractional parameters, which were previously unresolved in the literature.

The authors emphasize that their results rely on the interaction between the two distinct damping mechanisms and the specific regularity provided by the moduli of continuity, offering a refined perspective on the critical behavior of such evolution systems.