12 problems

Let $P_{k_0(\lambda)}$ and $Q_{k_0(\lambda)}$ denote the remainder terms in the time asymptotic expansion for the compressible Euler equations with time-dependent damping. All-orde…

Extension conjecture. This observation may also hold for equations similar to $$, including equations with Tricomi-type or mass terms and coupled wave systems.

Polynomial decay conjecture. The decay rate

Optimality conjecture. In view of the asymptotic behaviour of the eigenvalues of $\mathcal{A}$, the energy decay rate of type

Asymptotic-equivalence conjecture. It is conjectured that the system is time-asymptotically equivalent to the decoupled system

Let $w$ solve the equation denoted by … lambda=1,quad 0<muleq 2,qquadtext{or}qquad lambda1,quad mu0. … will blow up in finite time for a family of smooth initial data…

Square-root damping conjecture. The choice $\theta=1/2$, corresponding to square-root damping, is sufficient to obtain energy estimates resulting in well-posedness for the inextens…

Let $n$ be the spatial dimension, let $\mu$ be the damping parameter, and consider the small data Cauchy problem … for $(t,x)\in \mathbb{R}_+\times \mathbb{R}^n$. Here $p_c(n+\mu)$…

Generalized criticality conjecture. The parameter $\widetilde{\alpha}_{\max}(1,1/2)=3/2$ is critical when two exponents are below or equal to the exponent $p_c(1,1/2)$. This genera…

Consider the $N$-dimensional half-space analogue of the wave problem with damping coefficient $V(x)$ whose decay at spatial infinity is governed by a parameter $V_0$, and let…

Consider the one-dimensional half-line wave problem with damping coefficient $V(x)$ satisfying … and, for some $V_0>0$, $V_1\geq V_0$, and $L_2>0$, … Let $E(t)$ denote the energy o…

Let $T>\mu>0$, let $a<b$ with $(a,b)\subset(0,1)$, and let $\phi_n(x)=\sqrt{2}\sin(n\pi x)$. For each measurable set $\Xi\subset[0,T]$ of measure $\mu$, consider the largest consta…