69 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 $P$ be a critical operator in a Riemannian manifold $\mathcal{M}$, and let $\varphi$ denote its ground state. Let $\mathcal{H}_P(\mathcal{M}\times\mathbb{R})$ be the cone of po…

Let $F(u)=\frac{1}{4}(1-u^2)^2$, so that $F'(u)=u^3-u$, and consider the reaction-diffusion equation … in $\mathbb{R}^n\times[0,\infty)$. Suppose that $u$ satisfies $|u|\leq 1$ and…

Consider the generalized thin-film equation … with $a_0>0$, $a_1>0$, and nonnegative solutions subject to boundary conditions preserving the mass $\int h(x,t)\,dx$. The supercritic…

Matrix-valued Feynman–Kac representation conjecture. Under suitable regularity conditions, the solution of the system is

Slow blow-up existence conjecture. The analysis of He and Velázquez can be used to prove the existence of slow blow-up solutions for $n>4$.

Physical relevance conjecture. The solutions constructed by the numerical scheme are physically relevant.

Let $u_\alpha$ and $u_\beta$ be the corresponding solutions of the fractional parabolic equation … with $0<\beta<\alpha<1$ and non-negative, even, unimodal initial data as specifie…

Let $\Omega\subset\mathbb{R}^N$, with $1\leq N\leq 3$, be a bounded domain with smooth boundary, and let $u_0\in H^2\cap H_0^1(\Omega)$. Let $\lambda^{*}$ be the critical parameter…

Let $\Omega\subset\mathbb{R}^N$, with $1\leq N\leq 3$, be a bounded domain with smooth boundary, and let $u_0\in H^2\cap H_0^1(\Omega)$. Consider equation (1.1), its solution…

Let $p(x,z;t)$ be the Mehler kernel of the Ornstein–Uhlenbeck semigroup, defined for $t>0$ and $(x,z)\in\mathbb R^n\times\mathbb R^n$ by … The kernel $p$ is log-concave in $(x,z)$.…

CFG08 extension conjecture. The proof of the analogous blow-up assertion can be carried out without assuming $u(0,t)=\max_{\Omega}u(\cdot,t)$, provided that the conclusions of Lemm…

Let $u_0$ be initial data on $4\mathbb{R}^n4$, and let $4\alpha4$ and $p$ satisfy the hypotheses of problem (44). In particular, assume … The initial data are required to satisfy ……

Singular-measure existence conjecture. The same existence results should continue to hold when $\nu$ is singular with respect to $\mathcal{L}$.

Convergence conjecture. Alternative (ii) of Theorem 2.1 holds: $\|S(t,\cdot)-\tilde{S}\|_{\infty}\to0$ and $\|I(t,\cdot)-\tilde{I}\|_{\infty}\to0$ as $t\to\infty$. The proposition…

The paper develops invariant-manifold theorems for random parabolic evolution equations with almost sectorial operators, including local stable and center manifolds and conditions…

Consider sequences of parabolic nonlocal operators of the form … In the local setting, when the sequence of matrix-valued functions is independent of time, the parabolic $H$-limit…

Computational norm conjecture. The $H^2(Q_T)$-norms work well computationally in the numerical tests because all norms on a finite-dimensional space are equivalent and the relative…

Uniqueness conjecture. Uniqueness holds across the entire range

Extension conjecture. Theorem S should hold for every $n\geq 1$, and in part (ii) one should have

The paper studies boundary approximate controllability under positivity constraints for transport and heat systems. Applications conjecture. Other potential applications include mo…

Kinetic Krylov–Safonov conjecture. There exist constants $\alpha>0$ and $C$, depending only on the dimension and $\lambda,\Lambda$, such that

The polyharmonic heat equation is obtained by replacing the biharmonic operator with a higher-order polyharmonic operator, and its linear counterpart is the corresponding homogeneo…

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

Optimal decay estimates conjecture. These optimal estimates should remain valid when $a\neq b$.