Short-time existence and regularity for nonlinear heat equation

1. Review of Sobolev spaces and Linear equations.
2. Regularity estimate of the quadratic term.
3. Regularity for nonlinear heat equation.
4. Short-time existence for nonlinear heat equation.

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We will establish in this part a regularity estimate for the quadratic term of nonlinear heat operator use it to setup a bootstrap scheme that eventually will prove that any sufficiently regular solution of nonlinear heat equation that is initially $ C^\infty $ will be always $ C^\infty $.

We will also prove short-time existence using well-known method of Inverse function theorem for Banach spaces. Since the solution is smooth, we can apply Theorem thm:global-eq to conclude that the it remains in $ M'\subset \mathbb{R}^N $.

1 Review of Sobolev spaces and Linear equations.

The following results are well-known and their statements are written here in the case of our interest (linear heat equation on manifold). A more careful formulation with complete proofs can be found in the appendices.

1.1 Sobolev spaces.

Let $ M $ be a Riemannian manifold, the Sobolev spaces $ W^{k,p}(M) $ on $ M $ can be defined as the completion of $ C^\infty(M) $ with respect to the Sobolev norms \[ \|\varphi\|_{W^{k,p}} = \sum_{|\alpha|\leq k} \| D^\alpha \varphi \|_{L^p}. \] We will suppose that $ M $ is a compact manifold, then set-theoretically $ W^{k,p} $ does not depend on the metric of $ M $ and their norm remains in the same equivalent class as the metric varies. The Sobolev spaces form a family of reflexive Banach spaces that is stable under holomorphic interpolation:

Let $ p,q\in (1,+\infty) $ and $ k,l\in \mathbb{R} $ and $ M $ be a compact Riemannian manifold. Then the holomorphic interpolations of \[ A_0:= W^{k,p}(M)\quad \text{and}\quad A_1:=W^{l,q}(M) \] are $ A_\theta=W^{s,r}(M) $ where \[ \theta l + (1-\theta)k = s,\qquad \theta \frac{1}{q} + (1-\theta) \frac{1}{p} = \frac{1}{r}. \] In particular, one has the Interpolation inequality \[ \|f\|_{W^{s,r}}\leq 2 \|f \|^\theta_{W^{l,q}} \| f\|^{1-\theta}_{W^{k,p}}. \]

Sobolev embeddings and Kondrachov theorem remain correct on manifold.

Given $ k,l\in \mathbb{Z},\ k>l \geq 0 $ and $ p,q\in \mathbb{R},\ p>q\geq 1 $. Then

If $ \frac{1}{p}= \frac{1}{q} - \frac{k-l}{n} $ then \[ W^{k,q}(M) \hookrightarrow W^{l,p}(M), \]
If $ \frac{k-r}{n}> \frac{1}{q} $ then \[ W^{k,q}(M) \hookrightarrow C^r(M) \] If $ \frac{k-r-\alpha}{n}\leq \frac{1}{q} $ then \[ W^{k,q}(M) \hookrightarrow C^{r,\alpha}(M) \]

where $ C^r(M) $ denotes the space of $ C^r $ functions equipped with the norm $\| u \|_{C^r} = \max_{l\leq r}\sup|\nabla^l u|$, and $ C^{r,\alpha} $ is the subspace of $ C^r $ of functions whose $ r^{\rm th} $-derivative is $ \alpha $-Holder, equipped with the norm $ \| u\|_{C^{r,\alpha}} = \| u \|_{C^r} + \sup_{P\ne Q}\{ \frac{u(P) - u(Q)}{d(P,Q)^\alpha} \} $.

Let $ k\in \mathbb{Z}_{\geq 0} $ and $ p,q\in \mathbb{R}_{>0} $ be such that $ 1\geq \frac{1}{p} > \frac{1}{q} - \frac{k}{n} > 0$ then

The embedding $ W^{k,q}(M) \hookrightarrow L^p(M) $ is compact,
The embedding $ W^{k,q}(M) \hookrightarrow C^\alpha(M) $ is compact if $ k-\alpha > \frac{n}{q} $ where $ 0\leq \alpha < 1 $,

It is also natural, for regularity results of parabolic equation, to use weighted Sobolev spaces because each derivative in time should be counted as twice as that in space. For example, the space $ W^{2,p}(M\times[\alpha,\omega]) $ is the completion of $ C^\infty(M) $ with respect to the norm \[ \|\varphi\|_{W^{2,p}} := \|\varphi\|_{L^p} + \left\| \frac{d\varphi}{dt}\right\|_{L^p} + \sum_{i,j} \left\| \frac{\partial^2\varphi}{\partial x^i \partial x^j} \right\|_{L^p} + \sum_i \left\| \frac{\partial\varphi}{\partial x^i} \right\|_{L^p} \] Similarly, one can define $ W^{2k,p}(M\times[\alpha,\omega]) $ using $ L^p $-norm of derivatives $ \partial_t^\beta \partial_x^\gamma \varphi $ of $ \varphi $ with $ 2\beta + \gamma \leq 2k $.

We also want to be able to talk about $ W^{k,p} $ when $ k $ is not an integer and not necessarily positive. This allows us to have a more flexible bootstrap scheme for nonlinear heat equation and to use Interpolation Theorem thm:interp-sobolev-d more efficiently. We claim that these generalised Sobolev spaces (with weight and with non-integral regularity) can be defined on manifold and satisfy all the above properties (reflexivity, Interpolation theorem, Sobolev embedding and Kondrachov theorem) and refer to the appendices for all the details.

1.2 Trace theorem.

It is possible to avoid a discussion on Trace operator if we only want to make sense of the initial condition of nonlinear heat equation: one can consider only solutions with regularity greater than $ W^{2,p}(M\times[\alpha,\omega]) $ with $ p \geq \dim M + 2 $, which can be embedded in $ C(M) $. It is however necessary to investigate regularity of Trace operator to have a complete proof of the bootstrap. We will review briefly some results.

The following two behaviors of trace are well-known:

If $ -1 + \frac{1}{p} < k < \frac{1}{p} $ then the natural map $ W^{k,p}(M\times[\alpha,\omega]/\alpha) \hookrightarrow W^{k,p}(M\times[\alpha,\omega]) $ is an isomorphism, where $ W^{k,p}(M\times[\alpha,\omega]/\alpha) $ denotes the completion under $ W^{k,p} $-norm of the space of smooth functions vanishing on a neighborhood of $ M\times \{\alpha\} $. There is therefore no meaningful notion of trace in this case.
If $ k > \frac{1}{p} + l,\ l\geq 0$, then the restriction map \[ B:\ C^\infty(M\times[\alpha,\omega]) \longrightarrow C^\infty(M):\ f(x,t) \longmapsto f(x,\alpha) \] extends to a bounded operator $B:\ W^{k,p}(M\times[\alpha,\omega]) \longrightarrow W^{l,p}(M) $, called Trace operator.

We will topologise the space $ \partial_\alpha W^{k,p}(M\times[\alpha,\omega]) $ of restrictions to time $ t=\alpha $ of functions in $ W^{k,p}(M\times[\alpha,\omega]) $, in case Trace operator is well defined, as cokernel of $ B $, that is, as a quotient space of $ W^{k,p}(M\times[\alpha,\omega]) $. This makes $ \partial_\alpha W^{k,p}(M\times[\alpha,\omega]) $ a Banach space with stronger norm than any $ W^{l,p}(M) $ for any $ l < k-\frac{1}{p} $.

1.3 Linear equations on manifolds.

1.3.1 Existence and Regularity.

It can be easily verified that the linear heat operator $AF:= \frac{d}{dt}F + \Delta F $ is a parabolic operator and therefore is also an elliptic operator. All of the following results holds for operator $ A $.

Let $ M $ be a compact manifold and $ AF:= \frac{d}{dt}F + \Delta F $ be an elliptic operator of second order. Given $ \frac{1}{p} < l < k <\infty $ and $ F\in W^{l,p}(M\times[\alpha,\omega]) $ and suppose that \[ AF\in W^{k-2,p}(M\times[\alpha,\omega]),\quad \restr{f}{\alpha}\in \partial_\alpha W^{k,p}(M\times[\alpha,\omega]), \quad \restr{f}{\omega}\in \partial_\alpha W^{k,p}(M\times[\alpha,\omega]). \] Then actually $ F\in W^{k,p}(M\times[\alpha,\omega]) $.

Let $ M $ be a compact manifold and $ AF:= \frac{d}{dt}F + \Delta F + a\nabla F + bF $ be an parabolic operator. Then \[ A:\ W^{k,p}(M\times[\alpha,\omega]/\alpha) \longrightarrow W^{k-2,p}(M\times[\alpha,\omega]/\alpha) \] is an isomorphism of Banach spaces.

Let $ M $ be a compact manifold, $ p\in (1,+\infty) $, $ k > l > -\infty$ and $ AF:= \frac{d}{dt}F + \Delta F$ be a parabolic operator. We write $ W^{k,p}([\beta,\gamma]) $ shortly for $ W^{k,p}(M\times[\beta,\gamma]) $. Suppose that \[ F\in W^{l,p}([\alpha,\omega]),\quad AF\in W^{k-2,p}([\alpha,\omega]). \] Then $ F\in W^{k,p}([\pi,\omega]) $ for all $ \pi\in (\alpha,\omega) $. Also, there exists a constant $ C>0 $ such that \[ \|F\|_{W^{k,p}([\pi,\omega])} \leq C \left( \|AF\|_{W^{k-2,p}([\alpha,\omega])} + \| F \|_{W^{l,p}([\alpha,\pi])} \right). \] In particular for homogeneous equation, the solution is $ C^\infty $ and an arbitrarily weak estimate in the past gives an arbitrarily strong estimate in the future.

1.3.2 Maximum principle and Comparison theorems.

Other than regularity results which are generally true for parabolic operators, the linear heat operator also enjoys the following versions of Maximum principle. See Appendices for their proofs.

Let $ M $ be a compact manifold and $ f: M\times[\alpha,\omega] \longrightarrow \mathbb{R} $ be a continuous function with $ \restr{f}{\alpha}\leq 0 $. Suppose that whenever $ f>0 $, $ f $ is smooth and \[ \frac{\partial f}{\partial t} \leq -\Delta f + Cf. \] Then in fact $ f\leq 0 $.

With the same proof as Theorem thm:max-princ-d, one also has:

Let $ f: M\times[\alpha,\omega] \longrightarrow \mathbb{R} $ be a continuous function on $ M $, smooth for all time $ t>0 $ such that \[ \frac{d f}{dt} = -\Delta f + b f \text{ on } M\times (\alpha,\omega] \] where $ b $ is a smooth function on $ M $. Then there exists a constant $ B $ depending only on $ b $ such that \[ \|\restr{f}{\omega}\|_{L^\infty} \leq e^{B(\omega-\alpha)} \|\restr{f}{\alpha}\|_{L^\infty}. \]

Using backwards heat equation and Theorem thm:infty-comparison-d, one can prove its version for $ L^1 $.

Let $ f: M\times[\alpha,\omega] \longrightarrow \mathbb{R} $ be a continuous function on $ M $, smooth for all time $ t>0 $ such that \[ \frac{d f}{dt} = -\Delta f + b f \text{ on } M\times (\alpha,\omega] \] where $ b $ is a smooth function on $ M $. Then there exists a constant $ B $ depending only on $ b $ such that \[ \|\restr{f}{\omega}\|_{L^1} \leq e^{B(\omega-\alpha)} \|\restr{f}{\alpha}\|_{L^1}. \]

2 Regularity estimate of the quadratic term.

Let $ F:\ M\times [\alpha,\omega] \longrightarrow B\subset \mathbb{R}^N $ be in $ W^{s,q}(M\times[\alpha,\omega])\cap C(M\times [\alpha,\omega]) $ and \[ P F := g^{ij}\Gamma'^\alpha_{\beta\gamma}(F) F^\beta_i F^\gamma_j. \] Suppose that

\begin{equation} \label{eq:cond:thm:reg-poly-diff} r\geq 0,\quad p,q\in (1,\infty),\quad r+1 < s, \quad \frac{1}{p}> \frac{r+2}{s} \frac{1}{q}. \end{equation}

Then one has $ PF\in W^{r,p}(X) $ and \[ \|P F \|_{W^{r,p}} \leq C\left(1 + \|F\|_{W^{s,q}}\right)^{q/p}. \] where $ C $ is a constant independent of $ F $.

We will suppose here that $ r,s $ are even integers so that the $ W^{r,p} $ (respectively $ W^{s,q} $) norm of $ PF $ (respectively $ F $) can be written as sum of $ L^p $ (respectively $ L^q $) norms of its derivatives. Also, we will use chain rule freely to differentiate the term $ \Gamma'^\alpha_{\beta\gamma}(F) $ using weak derivatives of $ F $. The general and rigorous proof, which involves non-integral Sobolev space to treat $ r, s $ and a detour to Besov spaces to justify chain rule, can be found in the appendices.

The derivatives of $ PF $ that appear in its $ W^{r,p} $ norm are of form \[ C(x,F)\ \prod_i \partial_t^{b_i} \partial_x^{c_i} F^{\beta_i} \] where $ 2\sum b_i + \sum c_i \leq r+2 $ and $ \max \{2b_i+ c_i\}\leq r+1 $ and $ C(x,F) $ is bounded on $ M $. Using Multiplication theorem for $ L^p $-spaces, one has \[ \left\|C(x,F)\ \prod_i \partial_t^{b_i} \partial_x^{c_i} F^{\beta_i} \right\|_{L^p} \leq \|C(x,F)\|_{L^\infty} \prod_i \left\|\partial_t^{b_i} \partial_x^{c_i} F^{\beta_i}\right\|_{L^{p_i}} \leq \|C(x,F)\|_{L^\infty} \prod_i \left\|F\right\|_{W^{2b_i+c_i,p_i}} \] as long as we choose $ p_i\in (1,\infty) $ such that $ \frac{1}{p} \geq \sum \frac{1}{p_i} $. The strategy is to choose $ \frac{1}{p_i} $ big enough to have $ W^{s,q} \hookrightarrow W^{2b_i+c_i,p_i} $ in order to bound $\left\|F\right\|_{W^{2b_i+c_i,p_i}}$ by $ \left\|F\right\|_{W^{s,q}} $, then use the upper bound of $ 2b_i +c_i $ to justify that $ \frac{1}{p} > \frac{r+2}{s} \frac{1}{q} \geq \sum \frac{1}{p_i} $, meaning that such choice of $ p_i $ are valid.

One straightforward way to have a sufficient condition of $ p_i $ such that $ W^{s,q} \hookrightarrow W^{2b_i+c_i,p_i} $ is to use Sobolev embeddings. Another way is to use Interpolation inequality, by remarking that $ F\in W^{0,v} $ for all $ v\in (1,+\infty) $ and writing $ W^{2b_i+c_i, p_i} $ as an interpolation space of $ W^{s,q} $ and $ W^{0,v} $. It can be seen, by direct computation, that the sufficient condition for $ W^{s,q} \hookrightarrow W^{2b_i+c_i,p_i} $ is $ 2b_i+c_i < s $ and \[ 0 < \frac{1}{p_i} - \frac{2b_i+c_i}{s}\frac{1}{q} < 1 - \frac{2b_i + c_i}{s}. \] Choose $ \frac{1}{p_i} $ just a bit bigger than $ \frac{2b_i+c_i}{s}\frac{1}{q} $, one still has \[ \sum \frac{1}{p_i} \simeq \sum \frac{2b_i+c_i}{s}\frac{1}{q} \leq \frac{r+2}{s}\frac{1}{q} < \frac{1}{p}. \] The conclusion follows.

3 Regularity for nonlinear heat equation.

Let $ p>\dim M + 2 $, using the regularity estimate for the quadratic term, we now can prove:

Let $ F:\ M\times [\alpha,\omega] \longrightarrow B $ such that $ F\in W^{2,p}(M\times [\alpha,\omega]) $ and $ \frac{d F_t}{dt} = \tau(F_t) $, i.e. \[ \frac{d F^\alpha}{dt} = -\Delta F^\alpha + g^{ij}\Gamma'^\alpha_{\beta\gamma}(F) F^\beta_i F^\gamma_j \] and $ \restr{F}{M\times\{\alpha\}} $ is smooth. Then $ F $ is smooth on $ M\times [\alpha,\omega] $.

Note that since $ p > \dim M + 2 = \dim (M\times [\alpha,\omega])+1 $, if $ F\in W^{2,p}(M\times[\alpha,\omega]) $ then $ F $ and $ \frac{\partial F}{\partial x^i} $ are in $ C(M\times[\alpha,\omega]) $ by Sobolev embeddings. It makes sense then to talk about:

the restriction and boundary condition at time $ t=\alpha $ (in fact, by Trace theorem, $ p>1 $ is enough).
the pointwise condition $ F: M\times [\alpha,\omega] \longrightarrow B\subset V $.

We define the operators $ P F := g^{ij}\Gamma'^\alpha_{\beta\gamma}(F) F^\beta_i F^\gamma_j $ and $ A F := \frac{dF }{dt} + \Delta F $. We will abusively denote $ W^{k,p}(M\times [\beta,\gamma])$ by $ W^{k,p}([\beta,\gamma]) $. Our bootstrap scheme consists of 3 steps:

Prove that $ F\in W^{2,\tilde p}([\pi,\omega]) $ for every $ \pi > \alpha $ and $ \tilde p \in (1,\infty) $. By compactness of $ M $, it is sufficient to prove this for a sequence $ \tilde p \to +\infty $.
Prove that $ F $ is $ C^\infty $ for all time $ t >\alpha $.
Prove that $ F $ is $ C^\infty $ on $ M\times [\alpha,\omega] $.

Step 1. By Theorem thm:reg-quad, $ AF = PF \in W^{r,q}([\alpha,\omega]) $ whenever $ r<1 $ and $ \frac{1}{q} > (\frac{r}{2}+1)\frac{1}{p} $. Apply Gårding inequality, for all $ \pi >\alpha $, $ F\in W^{r+2,q}([\pi,\omega]) \subset W^{2,\tilde p}([\pi,\omega]) $ for $ \frac{1}{\tilde p} = \frac{1}{q} - \frac{r}{\dim M + 1} $. Choose $ \frac{1}{q} $ very close to $ (\frac{r}{2}+1)\frac{1}{p} $, one sees that the condition on $ \tilde p $ is $\frac{1}{\tilde p } > (\frac{r}{2} +1 ) \frac{1}{p} - \frac{r}{p-1}$, which will be satisfied if $ \frac{1}{\tilde p} > (1-\frac{r}{2}) \frac{1}{p}$, i.e. for all $ \tilde p < \frac{p}{1 -r/2} $. It remains to repeat this result to finish the first step. We will say $ F\in W^{2,*}([\pi,\omega]) $ for $ F\in W^{2,p}([\pi,\omega]) $ for all $ p\in (1,\infty) $.

Step 2. By Theorem thm:reg-quad, for all $ r<1 $, one has $ AF = PF \in W^{r,*}([\pi,\omega]) $, therefore by Gårding inequality, $ F\in W^{r+2, * }([\pi,\omega]) $. Iterate this result and one has $F\in W^{k,*}([\pi,\omega]) $ for all $ k\in [2,\infty) $ and $ \pi >\alpha $. So $ F $ is smooth for $ t>\alpha $.

Step 3. We apply regularity result (Theorem thm:elliptic-d) for elliptic operator $ A $ and boundary operators $ B^0: F \mapsto \restr{F}{M\times\{\alpha\}} $ and $ B^1: F \mapsto \restr{F}{M\times\{\omega\}} $: For $ q,r $ in Step 1, one has $ AF = PF \in W^{r,q}([\alpha,\omega]) $ and $ B^j F \in \partial W^{r,q} $, therefore $ F\in W^{r+2,q}([\alpha,\omega])\subset W^{2,\tilde p}([\alpha,\omega]) $ for the same $ \tilde p $ as Step 1. This proves that $ F\in W^{2,*}([\alpha,\omega]) $, which also means that one has $ F\in W^{r+2,q}([\alpha,\omega]) $ with no additional condition on $ q $ except $ q\in (1,\infty) $. Iterate and one obtains the regularity of $ F $ on $ [\alpha,\omega] $.

The first 2 steps were to prove the regularity of $ \restr{F}{M\times \{\omega\}} $, which was then used as a boundary condition in order to apply regularity result for elliptic operator on manifold with boundary.

4 Short-time existence for nonlinear heat equation.

We will choose as always $ p > \dim M +2 $. As before, $ M $ is a compact Riemannian manifold and $ B\subset \mathbb{R}^N $ is a large Euclidean ball.

Let $ F_\alpha: M \longrightarrow B$ be a smooth map, then there exist $ \epsilon>0 $ depending on $ F_\alpha $ and $ F: M\times [\alpha,\alpha + \epsilon] \longrightarrow B $ such that $ F\in W^{2,p}(M\times [\alpha,\alpha+\epsilon]) $ with $ \restr{F}{M\times \{\alpha\}} = F_\alpha $ and \[ \frac{d F_t}{dt} = \tau(F_t) \quad \text{on } M\times [\alpha,\alpha +\epsilon] \]

We find $ F $ as a sum $ F = F_b + F_\# $ where $ F_b\in C^\infty(M\times [\alpha,\omega]) $ satisfies the initial condition and $ F_\# \in W^{2,p}(M\times [\alpha,\alpha+\epsilon]/\alpha) $.

The nonlinear heat operator can be written as:

\begin{align*} T: W^{2,p}(M\times[\alpha,\omega]/\alpha)^{\oplus N} &\longrightarrow L^p(M\times[\alpha,\omega])^{\oplus N}\\ F_\# &\longmapsto \tau(F_b + F_\#) \end{align*}

where $ \tau(F)^\alpha = - \Delta F^\alpha + g^{ij}\Gamma'^\alpha_{\beta\gamma}(F) F^\beta_i F^\gamma_j $, which can be rewritten as $ \tau(F) = -\Delta F + \Gamma(F)(\nabla F)^2 $. The derivative of $ T $ at $ F_\# $ in direction $ k\in W^{2,p}(M\times[\alpha,\omega]/\alpha)^{\oplus N} $ is \[ DT(F_\#) k = -\Delta k +D \Gamma(F)\cdot k . (\nabla F)^2 + 2\Gamma(F) \nabla F. \nabla k, \] or in local coordinates: \[ DT(F_\#)^\alpha = g^{ij}\left( \frac{\partial^2 k^\alpha}{\partial x^i \partial x^j} - \Gamma^l_{ij} k^\alpha_l \right) + g^{ij} \frac{\partial \Gamma'^\alpha_{\beta\gamma}}{\partial y^\delta} k^\delta F^\beta_i F^\gamma_j + 2g^{ij} \Gamma'^\alpha_{\beta\gamma}(F) F^\beta_i F^\gamma_j \] which is of form $ DT(F_\#) k = -\Delta k - a(x,F) \nabla k -b(x,F) k $ where $ a,b $ are smooth.

Therefore if we note

\begin{align*} H: W^{2,p}(M\times[\alpha,\omega]/\alpha)^{\oplus N} &\longrightarrow L^p(M\times[\alpha,\omega])^{\oplus N}\\ F_\# &\longmapsto (\frac{d }{dt} - \tau)(F_b + F_\#) \end{align*}

then the derivative of $ H $ at $ F_\# = 0 $ is \[ DH(0)\cdot k = \frac{d k}{d t} + \Delta k + a(x, F_b) \nabla k + b(x, F_b) k \] which by Theorem thm:para-existence-d is an isomorphism from $ W^{2,p}(M\times[\alpha,\omega]/\alpha)^{\oplus N}$ to $W^{0,p}(M\times[\alpha,\omega]/\alpha)^{\oplus N} = L^p(M\times[\alpha,\omega])^{\oplus N} $. This shows that $ H $ is a local isomorphism mapping a neighborhood of $ 0 $ to a neighborhood of $ (\frac{d }{dt}-\tau)F_b $.

Define $ g_\epsilon\in L^p(M\times [\alpha,\omega])^{\oplus N} $ by \[ g_\epsilon:= \begin{cases} 0 , & \text{if $t\in[\alpha,\alpha+\epsilon]$} \\ (\frac{d }{dt}-\tau)F_b , & \text{if $t > \alpha+\epsilon$} \end{cases} \] which is arbitrarily $ L^p(M\times[\alpha,\omega]) $-close to $ (\frac{d }{dt}-\tau)F_b $ for $ 0<\epsilon \ll 1 $. There exists therefore $ F_\#\in W^{2,p}(M\times [\alpha,\omega]/\alpha)^{\oplus N} $ such that $ H(F_\#) = g_\epsilon $, meaning that the function $ F= F_b + F_\#: M \longrightarrow V $ satisfies $ \restr{F}{M\times\{\alpha\}} = F_\alpha $ and $ \frac{d F}{d t} -\tau(F_t) = 0 $ for $ t\in[\alpha,\alpha+\epsilon] $.

By Regularity Theorem thm:reg-nonlin-heat, $ F $ is $ C^\infty $ for $ t\in[\alpha,\alpha+\epsilon] $. Theorem thm:global-eq assures that the image of $ F $ is in $ M $, hence in $ M' $ for $ t\in [\alpha,\alpha+\epsilon] $.