28 problems

Glassey's critical-threshold conjecture. There should exist a global-in-time small-data solution when $C_{\mathrm{Gla}}=0$.

Let $u^I$ be a solution to a system of semilinear wave equations whose nonlinearities satisfy the weak null condition: the associated asymptotic system … has a global solution for…

Subcritical lifespan conjecture. If $1<p<p_0(n)$ with $n\geq3$, or $2<p<p_0(2)$ with $n=2$, then

For the $2$-D semilinear wave equation with scale-invariant damping … where $t\ge 1$, $\mu>0$, and $p>1$, a global small-data weak solution is a weak solution $u$ arising from suff…

Let $n\geq 4$, $\mu>0$, and consider the regular Cauchy problems … with small initial data, or … with small initial data. Denote by $p_S$ the Strauss exponent and by $p_F$ the Fuji…

Stability and quenching-profile conjecture. The following statements are conjectured:

Forbidden mass interval conjecture. The interval

Critical-power conjecture. For $\alpha=0$, the small-data Cauchy problem admits the critical power

Critical Strauss lifespan conjecture. The lifespan satisfies

Consider the semilinear regular Euler–Poisson–Darboux–Tricomi equation … bigl((m+1)n-1+mubigr)p^2-bigl((m+1)(n-2)+3+mubigr)p-2(m+1)=0. … . This conjecture proposes which of the Str…

Global well-posedness conjecture. Global well-posedness in the energy space holds throughout the range

Global well-posedness conjecture. For smooth radially symmetric initial data, global well-posedness holds provided that $0\leq b\leq 1/2$.

Integral criterion conjecture. Small-data global existence holds if

Chen–Reissig's conjecture. If $C_{\mathrm{Str}}>0$, then the solution blows up at finite time for some initial data; if $C_{\mathrm{Str}}=0$, then small-data global existence holds…

Consider wave models whose derivative-type nonlinearity has the form $|u_t|^{p_{\mathrm{crit}}(n)}\mu(|u_t|)$, where $p_{\mathrm{crit}}(n)$ is a possible critical exponent and…

Let $n$ be the spatial dimension, let $p_{\mathrm{Str}}(n)$ denote the Strauss exponent, and let $\mu$ be the modulus of continuity in a nonlinearity of the form…

Let $\mu$ be a modulus of continuity satisfying the global-existence condition … For the three-dimensional semilinear classical wave equation with radial small initial data, the st…

Shifted Strauss exponent conjecture. The critical exponent could be

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 $(u_0,u_1)$ be nontrivial, nonnegative, compactly supported smooth initial data for the semilinear wave equation on an asymptotically Euclidean manifold, with small-data size ……

Consider the defocusing semilinear wave equation in spatial dimension $d$, with nonlinearity power $p$, and the scattering result discussed in the source. Sharp lower-bound conject…

Constant-bound conjecture. The right-hand sides in the inequalities of Theorem and Corollary should be constant rather than powers of the logarithmic time variable; equivalently, t…

Critical-line conjecture. The critical line in the $(p,q)$-plane is given by

Consider the semilinear wave equation … in space dimension $N$, with $V_0>V_*$, where $V_*$ is the threshold value discussed in the source. Let $p_F(1)=1+\frac{2}{N-1}$ be the Fuji…

Consider the linear damped wave equation … with compactly supported smooth initial data $u^0(x,0)=u_1(x)$ and $u^0_t(x,0)=u_2(x)$ on $\mathbb{R}^n$. The damping is called non-effec…