At an interior maximum, $\dot{E}(t)=0$, equivalentlyThusThis is positive for distinct positive rates. Its continuous limit when $q_{EI}=q_{SE}=q$ is $1/q$.

Substitute part (ii) into $\dot{E}+q_{EI}E=q_{SE}{S}_0{e}^{-q_{SE}t}$. Multiplication by the integrating factor $e^{q_{EI}t}$ givesSince $E(0)=0$, integration and rearrangement yield

The separable equation $\dot{S}=-q_{SE}S$ with $S(0)=S_0$ has solution

The linear SEIR model with a constant susceptible-to-exposed intensity isAdding the four equations confirms conservation of population: $S+E+I+R=N$.

If $[M{]}_T\le b$, the Novikov condition holds for $(M-M_0)/2$ becauseIts stochastic exponential is therefore a true martingale with expectation one. Using the factorization from part (i),

Factor the square root of the stochastic exponential asThe stochastic exponential in this expression is a positive local martingale and hence a supermartingale, so its expectation is at most one. If $[M{]}_T\ge a$ almost surely, then

Reversing the finite summations givesThis is a Riemann sum for the continuous functional $f\mapsto {\int }_0^{1}f(t)dt$ evaluated at the interpolated random walk; the interpolation error tends to zero in probability. The continuous mapping theorem and part (b) yield

The evaluation map $f\mapsto f(1)$ is continuous on $C[0,1]$ in the uniform norm. The continuous mapping theorem applied to part (c) therefore givesThis recovers the central limit theorem from the functional version.

Change variables $x={\lambda }_{n}y$ and use the profiles from part 5:Since $v_n\to Q{e}^{i\gamma }$ in $L^2$, the difference made by replacing $|v_n{|}^2$ with $|Q{|}^2$ tends to zero against the bounded function $\varphi ({\lambda }_n\cdot )$. For each fixed $y$, continuity gives $\varphi ({\lambda }_{n}y)\to \varphi (0)$, and dominated convergence then givesThus the mass measures $|u(t_n,x){|}^{2}dx$ converge weakly to the Dirac delta function $\Vert Q{\Vert }_2^2{\delta }_0$.

Choose a smooth radial cutoff ${\theta }_R$ that vanishes on $|x|\le R$, equals one on $|x|\ge 2R$, and satisfies $|\nabla {\theta }_R|\le C/R$. The first localized virial identity and the estimate from part 2, applied to $w=u(t)$ and $\psi ={\theta }_R$, givebecause the mass and energy are conserved.

For fixed $R$, the strong profile convergence from part 5 and ${\lambda }_n\to 0$ implyIntegrating the flux estimate from $t$ to $t_n$ and then letting $n\to \infty$ yieldsuniformly for $0\le t<T$. Taking $R$ sufficiently large proves the claim.

By the Blowup alternative for the nonlinear Schrödinger equation, choose $t_n\uparrow T$ with $\Vert \nabla u(t_n){\Vert }_2\to \infty$, and setThe mass-critical scaling gives

The sequence is bounded and radial in $H^1$, so part 4 yields, after extraction, weak $H^1$ and strong $L^4$ convergence to some $v$. The energy identity gives $\Vert v_n{\Vert }_4^4\to 2\Vert \nabla Q{\Vert }_2^2$. The Sharp Gagliardo-Nirenberg inequality and weak lower semicontinuity force equality in every limiting norm inequality. Hence $v_n\to v$ strongly in $H^1$, and the variational characterization identifiesfor some $\gamma \in \mathbb{R}$; radiality removes translations and the gradient normalization fixes the scale.

Let $(u_n)$ be bounded in $H_r^1({\mathbb{R}}^2)$. After passing to a subsequence it converges weakly in $H^1$, while the Rellich-Kondrachov compactness theorem gives strong $L^4$ convergence on every bounded ball. The Radial Sobolev inequality gives uniformlyThe same estimate applies to the weak limit. Choosing $R$ large and then using local compactness proves strong convergence in $L^4({\mathbb{R}}^2)$. Henceis compact.

Suppose that $T=\infty$. Since $s_c<1$ and $d\ge 3$,The Radial Sobolev inequality and Mass conservation for the nonlinear Schrödinger equation giveBecause $(p-1)/2<2$, Young inequality and a sufficiently large fixed $R$ absorb this term into the left side of the estimate from part 5. The annular term is at most $C(u_0)R^{-2}$. Using the uniform lower bound $\Vert \nabla u(t){\Vert }_2\ge c(u_0)$ from part 2 and enlarging $R$ once more yields

Since $I_{{\psi }_R}^{\prime }=2V_{{\psi }_R}$, two integrations giveThe right-hand side is negative for large $t$, contradicting $I_{{\psi }_R}(t)=\int {\psi }_R|u{|}^2\ge 0$. Hence the maximal forward lifespan is finite: $T<\infty$.

For ${\psi }_R(x)=R^2\psi (x/R)$, the assumptions imply $\mathrm{Hess}{\psi }_R\le I$, while on $|x|\le 2R$ one hasApply the localized virial identity and compare its interior terms withThe coefficient $d(p-1)/4$ is strictly greater than one because $s_c>0$. The resulting negative multiple of $\Vert \nabla u{\Vert }_2^2$ can be moved to the left. All errors in the nonlinear term are supported on $|x|\ge 2R$, and ${\Delta }^2{\psi }_R=O(R^{-2})$ is supported on $2R\le |x|\le 10R$. Thus

For a smooth radial function that decays at infinity,The Cauchy-Schwarz inequality therefore yields the Radial Sobolev inequalityup to the harmless common normalization of surface measure. Taking the supremum over $r\ge R$ gives the claimed estimate; density extends it from smooth radial functions to every $u\in H_r^1$.

SetSubstituting $u_t=i(\Delta u+|u{|}^{p-1}u)$ and integrating by parts gives the localized virial identityDifferentiating once more, integrating the Laplacian terms twice, and observing that the gauge-invariant nonlinearity contributes only through $\Delta \chi$, givesThese are the required formulas, with ${\chi }^{\prime \prime }|\nabla u{|}^2$ interpreted as the Hessian quadratic form for a radial weight.

Take the regular radial solution with $u(0)=1$. In the transformed variables it satisfiesso $J(s)\to 0$. Since $s_c>1$, the Lyapunov function immediately becomes negative. The trajectory cannot reach $v=0$, where $J={\dot{v}}^2\ge 0$, and $J<0$ confines it below the positive zero of $f$. Hence it remains positive and bounded for all $s$.

The identity $\dot{J}=-4(s_c-1){\dot{v}}^2$ and the LaSalle invariance principle force the omega-limit set to consist of equilibria. The negative limiting energy excludes $v=0$, leavingConsequentlyso $\alpha =2/(p-1)$. At infinity, $|u^{\prime }(r)|\sim a{c}_{\infty }{r}^{-a-1}$, andbecause $s_c>1$. Thus $\nabla u\notin L^2({\mathbb{R}}^d)$.

Let $A=c_{\infty }^{p-1}$ andUsing the transformed equation,Thus $J$ is a strict Lyapunov function away from equilibria.

On ${\mathbb{R}}_{+}$, $f(0)=0$, it decreases to its unique minimum at $v=c_{\infty }$, wherethen increases, crosses zero at $v=((p+1)/2{)}^{1/(p-1)}{c}_{\infty }$, and tends to $+\infty$.

When $s_c=1$, one has $p=(d+2)/(d-2)$. The normalized Aubin-Talenti bubblesolves the radial equation, has $u(0)=1$, and tends to zero. It also follows from the stated one-dimensional soliton after the Emden-Fowler transformation, because the damping term vanishes.

As $r\to \infty$,so $\alpha =d-2$. Moreover $u^{\prime }(r)=O(r^{-(d-1)})$ at infinity and $u^{\prime }(r)=O(r)$ near zero. Thereforeand $\nabla u\in L^2({\mathbb{R}}^d)$.

Seek a homogeneous solution $u(r)=c{r}^{-\beta }$. Matching the powers in $\Delta u+u^p=0$ gives $\beta p=\beta +2$, henceSincematching coefficients givesThus the positive Singular homogeneous solution of the Lane-Emden equation is