23 problems

Let $u(x,t)$ be a solution with bounded growth of … where $W$ and $V$ are bounded, and $A(x,t)$ is uniformly elliptic and bounded. Landis–Oleinik conjecture. Under suitable conditi…

Let $u$ be a multi-soliton solution of the generalized KdV equation, defined at a given time, together with its spatial derivatives. Pointwise exponential decay conjecture. For eac…

Let $K=2,3,\dots$ and consider the initial-boundary value problem for a coupled subdiffusion system with $K$ components: … Assume $1\ge\alpha_1\ge\cdots\ge\alpha_K>0$, $\alpha_K<1$…

Consider the three-component coupled subdiffusion system in the case $v_0=w_0\equiv0$ and $u_0\not\equiv0$, with fractional orders $\alpha,\beta,\gamma$. In the numerical examples,…

Consider a coupled subdiffusion system with components having fractional derivative orders, and suppose some initial values may vanish identically. The lowest active fractional ord…

Let $w(t)$ be the global solution considered above, satisfying the decay estimates in particular for $\|w(t)\|_\infty$ as $t\to\infty$. Optimal $L^\infty$ decay conjecture. One wou…

Let $\psi^{\mathrm{dat}}$ be data as in the paper's scattering-data definition, with $d>0$ and $N>5$, for the linearised equation on the hypersurface $\Sigma_1$. Let…

Consider the defocusing nonlinear Schrödinger initial value problem … Let $s_c$ denote the critical regularity of the model, let $u^+$ be the forward scattering state, and assume t…

Consider the fast diffusion flow associated with the Caffarelli–Kohn–Nirenberg inequality, and let $T$ be the waiting time appearing in Theorem. The improved decay inequality is de…

Decay conjecture in the hyperbolic case.

Decay conjecture in the flat case.

Higher-order decay conjecture. The optimal rates in $$ should hold for all bounded and nonnegative solutions of the system.

Consider the defocusing semilinear wave equation on $\mathbb{R}^{1+2}$ with power $p$, for sufficiently smooth and localized initial data. Sharp-decay conjecture. If $p>3$, the sol…

Let $\mathcal{H}^+$ be the event horizon and let $v$ be the standard advanced time null coordinate. Massive scalar-field tail conjecture. For smooth, regular, generic admissible da…

Charged massless scalar decay conjecture. For smooth, regular, generic admissible data with a non-empty black hole and $m^2=0$,

Hod–Piran sharp decay conjecture. In the charged massless case, the scalar field is expected to satisfy

Let $b=b(t)$ be a general dissipation term satisfying … The previously established estimate for the corresponding Cauchy problem is denoted by. Derivative-free decay conjecture. Es…

Hod–Piran conjecture. Among all admissible, sufficiently regular and decaying data, there exists a generic subclass such that, whenever the maximal future development satisfies…

Consider the wave equation $\square_g\phi=0$ on de Sitter spacetime with flat $3$-dimensional spatial sections. Let $t$ denote cosmological time, let $a(t)$ be the scale factor, an…

Spencer's conjecture. The optimal decay rate in this estimate should be $-3d$.

Let $u(t,x)$ be the solution considered in the paper, with initial datum $u_0$, parameter $\mu$, and $t>1$. The notation $\\|u(t,\cdot)\\|_{L^{\infty}(\mathbb{R})}$ denotes the sup…

Let $u$ be a solution of the static Schrödinger–Poisson–Slater problem, and suppose that in the radial case its decay estimate is … for suitable constants $C>0$ and $R'>0$. Nonradi…

Decay conjecture. For every $N>0$, one can construct such a potential $u$ and eigenfunctions $\psi_1$ and $\psi_2$ that decay faster than $1/r^N$.