The Cauchy problem for a combustion model in a porous medium with two layers

Abstract

We prove the existence of local and global in time solutions of the Cauchy problem for a combustion model in a porous medium with two layers. The model is a system of four equations, consisting of two nonlinear reaction–convection–diffusion equations coupled with two ordinary differential equations, with the coupling occurring in both the reaction functions and in the differential operator coefficients. To obtain the local solution, we first construct an iteration scheme of approximate solutions to the system. Using the continuous dependence of solutions for parabolic equations with respect to the coefficients of the equations, we show that the constructed iteration scheme contains a sequence which converges to a local solution of the system, under the assumption that the initial data are Lipschitz continuous, bounded and non negative. We show that this solution can be extended globally, if the initial data are additionally in the Lebesgue space $L^p$, for some $p\in (1,\infty )$.

Appendix: Proof of Proposition 1

In this Section, we show the continuous dependence of the fundamental solutions for parabolic equations with respect to the coefficients of the equations and, as a consequence, we show Proposition 1 stated in Sect. 2. We use the notation fixed previously and the letters C and K shall denote positive constants that might depend on the parameters $R,\lambda ,\alpha ,T$, but not on the coefficients v neither on the solutions u or the data f and $u_0$ of (16), unless otherwise stated. In addition, K shall depend continuously on T.

Let us recall that the fundamental solution ${\varGamma }\equiv {\varGamma }_[v]$ is given by (see e.g. [7] or [9])

$$\begin{aligned} {\varGamma }(x,t,\xi ,\tau )= Z(x,t,\xi ,\tau )+\int _\tau ^t\int _{\mathbb {R}}Z(x,t,y,\sigma )\phi (y,\sigma ,\xi ,\tau )dyd\sigma \,, \end{aligned}$$

(37)

where $(x,t),\,(\xi ,\tau )\in {\varOmega }_T$, $t>\tau$, the function $Z(x,t,\xi ,\tau )$, as a function (x, t), is the fundamental solution of the heat equation $\frac{\partial u}{\partial t}-a(\xi ,\tau )\frac{\partial ^2 u}{\partial x^2}=0$, i.e.

$$\begin{aligned} Z(x,t,\xi ,\tau )=\frac{1}{(4\pi a(\xi ,\tau )(t-\tau ))^\frac{1}{2}}\text{ e }^{-\frac{(x-\xi )^2}{4a(\xi ,\tau )(t-\tau )}}, \end{aligned}$$

(38)

for each fixed $(\xi ,\tau )\in {\varOmega }_T$, and

$$\begin{aligned} \phi (x,t,\xi ,\tau )=\sum ^{\infty }_{m=1}(-\,1)^m(\mathcal {L}Z)_m(x,t,\xi ,\tau ), \end{aligned}$$

(39)

where $(\mathcal {L}Z)_1=\mathcal {L}Z=(a(\xi ,\tau )-a(x,t))\frac{\partial ^2 Z}{\partial x^2}+b\frac{\partial Z}{\partial x}+cZ$ and, for $m\ge 1$,

$$\begin{aligned} (\mathcal {L}Z)_{m+1}(x,t,\xi ,\tau )= \int _{\tau }^t\int _{\mathbb {R}}[\mathcal {L}Z(x,t,y,\sigma )](\mathcal {L}Z)_m(y,\sigma ,\xi ,\tau )dyd\sigma . \end{aligned}$$

(40)

We use very often the estimate

$$\begin{aligned} |D_t^rD_x^s{\varGamma }(x,t,\xi ,\tau )|\le \frac{K}{(t-\tau )^{\frac{1+2r+s}{2}}}\text{ e }^{-C\frac{(x-\xi )^2}{t-\tau }}, \end{aligned}$$

(41)

which holds for all $r,s\in \mathbb {Z}_{+}$ such that $2r+s\le 2$.

In the sequel we shall write $Z=Z_[v]$ and $\phi =\phi _[v]$. Then we have the following lemmas.

Lemma 7

Given $v=(a,b,c),\overline{v}=(\overline{a},\overline{b},\overline{c})\in B(R,\lambda ,\alpha )$, the following estimate is true.

$$\begin{aligned} |\left( D_x^sZ_{[v]}-D_x^sZ_{[\overline{v}]}\right) (x,t,\xi ,\tau )|\le K{\Vert a-\overline{a}\Vert }_\infty \frac{1}{(t-\tau )^ {\frac{s+1}{2}}}\text{ e }^{-C\frac{(x-\xi )^2}{(t-\tau )}}, \end{aligned}$$

for $s=0,1,2$, where $C<1/(4R)$ and $K=K(R,\lambda )$.

Proof

Since $Z_{[v]}(x,t,\xi ,\tau )=\frac{1}{(4\pi (t-\tau ))^{\frac{1}{2}}}\text{ e }^{-\frac{(x-\xi )^2}{4a(\xi ,\tau )(t-\tau )}}$ and its derivatives on x depends on the coefficient a of $\mathcal {L}_{[v]}$, but not depends on the other coefficients b and c, computing the derivative of $D_x^sZ_{[v]}$ with respect to a, we find $|D_aD_x^sZ_{[v]}(x,t,\xi ,\tau )|\le \frac{K}{(t-\tau )^{\frac{s+1}{2}}}\text{ e }^{-C\frac{(x-\xi )^2}{(t-\tau )}}$, for $s=0,1,2$, and constants K and C as in statement of the lemma. Then the desired inequality follows by the Mean Value Theorem. $\square$

The lemma below will help to show the next one.

Lemma 8

Let A and $\alpha$ be strictily positive numbers being $\alpha \le 1$, and let $\gamma$ denote the gamma function, i.e. $\gamma (x):=\int _0^\infty t^{x-1}\text{ e }^{-t}dt$. Then the series $\sum _{m=1}^\infty {mA^m}/{\gamma (\frac{m\alpha }{2})}$ is convergent.

Proof

We begin by recalling the relation $\frac{\gamma (x)\gamma (y)}{\gamma (x+y)}=\beta (x,y)$ between the gamma function $\gamma$ and the beta function, $\beta (x,y)=\int _0^1t^{x-1}(1-t)^{y-1}dt$ (see [13, p.41]). Denoting the general term of the given series by $a_m$, and using the above relation, we obtain

$$\begin{aligned} \lim _{m\rightarrow \infty }\frac{a_{m+1}}{a_m}= A\lim _{m\rightarrow \infty }\frac{\beta \left( \frac{m\alpha }{2},\frac{\alpha }{2}\right) }{\gamma \left( \frac{\alpha }{2}\right) } =A\lim _{m\rightarrow \infty }\frac{1}{\gamma \left( \frac{\alpha }{2}\right) }\int _0^1t^{\frac{m\alpha }{2}-1}(1-t)^{ \frac{\alpha }{2}-1}dt=0. \end{aligned}$$

Therefore, the result follows. $\square$

Lemma 9

Let $\beta \in [0,1]$ and $\gamma \in (0,\alpha )$. If $v,\overline{v}\in B(R,\lambda ,\alpha )$ then

$$\begin{aligned} |(\phi _{[v]}-\phi _{[\overline{v}]})(x,t,\xi ,\tau )|\le \frac{K\Vert v-\overline{v}\Vert _{\alpha ,\frac{\alpha }{2}}}{(t-\tau )^{\frac{3-\alpha }{2}}}\text{ e }^{-C\frac{(x-\xi )^2}{t-\tau }} \end{aligned}$$

(42)

and

$$\begin{aligned}&|(\phi _{[v]}(x,t,\xi ,\tau )-\phi _{[\overline{v}]}(x,t,\xi ,\tau ))- (\phi _{[v]}(y,t,\xi ,\tau )-\phi _{[\overline{v}]}(y,t,\xi ,\tau ))| \nonumber \\&\qquad \le \frac{K\Vert v-\overline{v}\Vert ^\beta _{\alpha ,\frac{\alpha }{2}}|x-y|^{\gamma (1-\beta )}}{(t-\tau )^ {\frac{3-(\alpha -\gamma (1-\beta ))}{2}}}\left( \text{ e }^{- C\frac{(x-\xi )^2}{t-\tau }}+ \text{ e }^{- C\frac{(y-\xi )^2}{t-\tau }}\right) , \end{aligned}$$

(43)

where $C<\frac{1}{4R}$ and $K=K(R,\lambda ,\alpha ,T)$.

Proof

The proof of (42) follows from the following inequality,

$$\begin{aligned}&|({(\mathcal {L}Z_{[v]})}_m-{(\mathcal {L}Z_{[\overline{v}]})}_m)(x,t,\xi ,\tau )| \end{aligned}$$

(44)

$$\begin{aligned}&\quad \le mK^m\left( \frac{\pi }{C}\right) ^{\frac{m-1}{2}}\Vert v-\overline{v}\Vert _{\alpha ,\frac{\alpha }{2}} \frac{g(\frac{\alpha }{2})^m}{g(\frac{m\alpha }{2})}\frac{1}{{(t-\tau )}^{\frac{3-m\alpha }{2}}}\text{ e }^{-C\frac{(x-\xi )^2}{t-\tau }}, \end{aligned}$$

(45)

where, for simplicity, we set $\mathcal {L}\equiv \mathcal {L}_{[v]}$, and g denotes the gamma function; see Lemma 8. Inequality (44) can be proved by induction on m, we skip the proof. $\square$

Lemma 10

Let $v,\overline{v}\in B(R,\lambda ,\alpha )$, $\beta \in (0,1)$, $\gamma \in (0,\alpha )$, and ${\varGamma }_{[v]}$. Then we have the following estimates:

$$\begin{aligned} |(D_x^s{\varGamma }_{[v]}-D_x^s{\varGamma }_{[\overline{v}]})(x,t,\xi ,\tau )|\le \frac{K\Vert v-\overline{v}\Vert _{\alpha ,\frac{\alpha }{2}}}{(t-\tau )^{\frac{s+1}{2}}}\text{ e }^{-C\frac{(x-\xi )^2}{t-\tau }},\quad s=0,1, \end{aligned}$$

(46)

$$\begin{aligned}&|\left( \partial _{xx}{\varGamma }_{[v]}-\partial _{xx}{\varGamma }_{[\overline{v}]}\right) (x,t,\xi ,\tau )| \le K\left( \Vert v-\overline{v}\Vert _{\alpha ,\frac{\alpha }{2}}+\Vert v-\overline{v}\Vert ^\beta _{\alpha ,\frac{\alpha }{2}}\right) \nonumber \\&\qquad \cdot \, \left( \frac{1}{|x-\xi |^{1-(\alpha -\gamma (1-\beta ))}(t-\tau )^{1-\frac{\gamma (1-\beta )}{2}}} +\frac{1}{(t-\tau )^{\frac{3}{2}}}\right) \text{ e }^{-C\frac{(x-\xi )^2}{t-\tau }} \end{aligned}$$

(47)

and

$$\begin{aligned}&|(\partial _{t}{\varGamma }_{[v]}-\partial _{t}{\varGamma }_{[\overline{v}]})(x,t,\xi ,\tau )| \le K\left( \Vert v-\overline{v}\Vert _{\alpha ,\frac{\alpha }{2}}+\Vert v-\overline{v}\Vert ^\beta _{\alpha ,\frac{\alpha }{2}}\right) \nonumber \\&\quad \cdot \, \left( \frac{1}{|x-\xi |^{1-(\alpha -\gamma (1-\beta ))}(t-\tau )^{1-\frac{\gamma (1-\beta )}{2}}} +\frac{1}{(t-\tau )^{\frac{3}{2}}}\right) \text{ e }^{-C\frac{(x-\xi )^2}{t-\tau }}, \end{aligned}$$

(48)

where $C<\frac{1}{4R}$ and $K=K(R,\lambda ,\alpha ,T)$.

Proof

The proofs of inequalities (46) and (47) follow from Lemmas 7, 8 and 9, and the proof of inequality (48) follows from (46) and (47). Here, we omit details of the proofs. $\square$

Next, we prove the Proposition 1.

Proof of Proposition 1

Writing

$$\begin{aligned} (u-\overline{u})(x,t)&=\int _\mathbb {R}{\varGamma }_{[v]}(x,t,\xi ,0)u_0(\xi )-{\varGamma }_{[\overline{v}]}(x,t,\xi ,0)\overline{u}_0(\xi )\,d\xi \nonumber \\&\quad + \int _0^t\int _\mathbb {R}{\varGamma }_{[v]}(x,t,\xi ,\tau )f(\xi ,\tau )-{\varGamma }_{[\overline{v}]}(x,t,\xi ,\tau )\overline{f}(\xi ,\tau )\,d\xi \,d\tau \\&\equiv V(x,t)+W(x,t), \end{aligned}$$

by standard estimates on the fundamental solutions, we obtain

$$\begin{aligned} |V(x,t)|&\le \int _\mathbb {R}|{\varGamma }_{[v]}(x,t,\xi ,0)u_0(\xi )-{\varGamma }_{[\overline{v}]}(x,t,\xi ,0)\overline{u}_0(\xi )|\,d\xi \nonumber \\&\le \int _\mathbb {R}|({\varGamma }_{[v]}-{\varGamma }_{[\overline{v}]})(x,t,\xi ,0)u_0(\xi )|+ |{\varGamma }_{[\overline{v}]}(x,t,\xi ,0)(u_0(\xi )-\overline{u}_0(\xi ))|\,d\xi \nonumber \\&\le \int _\mathbb {R}\frac{K\Vert v-\overline{v}\Vert _{1,\frac{1}{2}}}{t^{\frac{1}{2}}}\text{ e }^ {-C\frac{(x-\xi )^2}{t}}\Vert u_0\Vert _\infty +\frac{K}{t^{\frac{1}{2}}}\text{ e }^ {-C\frac{(x-\xi )^2}{t}}\Vert u_0-\overline{u}_0\Vert _\infty \,d\xi \nonumber \\&\le K\left( \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}+ \Vert u_0-\overline{u}_0\Vert _\infty \right) , \end{aligned}$$

(49)

where $K=K(R,\lambda ,T,\Vert u_0\Vert _\infty )$. In addition, using that the integral of derivatives of the fundamental solution is null, we can write

$$\begin{aligned} \partial _xV(x,t)&= \int _\mathbb {R}\partial _x{\varGamma }_{[v]}(x,t,\xi ,0)u_0(\xi )-\partial _x{\varGamma }_{[\overline{v}]}(x,t,\xi ,0)\overline{u}_0(\xi )\,d\xi \\&=\int _\mathbb {R}(\partial _x{\varGamma }_{[v]}-\partial _x{\varGamma }_{[\overline{v}]})(x,t,\xi ,0)(u_0(\xi )-u_0(x))\,d\xi \\&\quad +\int \partial _x{\varGamma }_{[\overline{v}]}(x,t,\xi ,0)[(u_0(\xi )-\overline{u}_0(\xi ))-(u_0(x)-\overline{u}_0(x))]\,d\xi , \end{aligned}$$

so, by Lemma 10 and estimate (41) and using that $\frac{|x-\xi |}{t} \text{ e }^{-C\frac{(x-\xi )^2}{t}}= \frac{1}{t^{1/2}} (\frac{|x-\xi |}{t^{1/2}} \text{ e }^{-(C/2)\frac{(x-\xi )^2}{t}})\text{ e }^{-(C/2)\frac{(x-\xi )^2}{t}} \le \text{ const }$. $\frac{1}{t^{1/2}}\text{ e }^{-(C/2)\frac{(x-\xi )^2}{t}}$, we obtain

$$\begin{aligned}&|\partial _xV(x,t)| \nonumber \\&\qquad \le \int _\mathbb {R}\left( \frac{K\Vert v-\overline{v}\Vert _{1,\frac{1}{2}} \Vert u_0\Vert _1|x-\xi |}{t}+ \frac{K\Vert u_0-\overline{u}_0\Vert _1|x-\xi |}{t}\right) \text{ e }^{-C\frac{(x-\xi )^2}{t}}\,d\xi \nonumber \\&\qquad \le \int _\mathbb {R}\frac{K\Vert u_0\Vert _1\Vert v-\overline{v}\Vert _{1,\frac{1}{2}}}{t^{\frac{1}{2}}}+ \frac{K\Vert u_0-\overline{u}_0\Vert _1}{t^{\frac{1}{2}}})\text{ e }^{-C\frac{(x-\xi )^2}{t}}\,d\xi \nonumber \\&\qquad \le K\left( \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}+\Vert u_0-\overline{u}_0\Vert _1\right) , \end{aligned}$$

(50)

with $K=K(R,\lambda ,T,\Vert u_0\Vert _1)$. To obtain the Hölder continuity with respect to t, we write

$$\begin{aligned}&V(x,t)-V(x,t')\\&\quad =\int _\mathbb {R}({\varGamma }_{[v]}(x,t,\xi ,0)-{\varGamma }_{[v]}(x,t',\xi ,0))u_0(\xi )\,d\xi \\&\qquad \int _\mathbb {R}-({\varGamma }_{[\overline{v}]}(x,t,\xi ,0)- {\varGamma }_{[\overline{v}]}(x,t',\xi ,0))\overline{u}_0(\xi )\,d\xi \\&\quad =\int _\mathbb {R}\int _{t'}^t\partial _t{\varGamma }_{[v]}(x,s,\xi ,0)u_0(\xi )- \partial _t{\varGamma }_{[\overline{v}]}(x,s,\xi ,0)\overline{u}_0(\xi ) \,ds\,d\xi \\&\quad =\int _{t'}^t\int _\mathbb {R}(\partial _t{\varGamma }_{[v]}-\partial _t{\varGamma }_{[\overline{v}]})(x,s,\xi ,0)u_0(\xi )+ \partial _t{\varGamma }_{[\overline{v}]}(x,s,\xi ,0)(u_0-\overline{u}_0)(\xi ) \,d\xi \,ds\\&\quad =\int _{t'}^t\int _\mathbb {R}(\partial _t{\varGamma }_{[v]}-\partial _t{\varGamma }_{[\overline{v}]})(x,s,\xi ,0)(u_0(\xi )-u_0(x)) \,d\xi \,ds\\&\quad \quad +\,\int _{t'}^t\int _\mathbb {R}\partial _t{\varGamma }_{[\overline{v}]}(x,s,\xi ,0)[(u_0-\overline{u}_0)(\xi )-(u_0-\overline{u}_0)(x)] \,d\xi \,ds. \end{aligned}$$

Therefore, from Lemma 10 and estimate (41), we obtain

$$\begin{aligned}&|V(x,t)-V(x,t')| \nonumber \\&\quad \le \int _{t'}^t\int _\mathbb {R}|\partial _t{\varGamma }_{[v]}-\partial _t{\varGamma }_{[\overline{v}]})(x,s,\xi ,0)||u_0(\xi )-u_0(x)| \,d\xi \,ds \nonumber \\&\quad \quad +\int _{t'}^t\int _\mathbb {R}|\partial _t{\varGamma }_{[\overline{v}]}(x,s,\xi ,0)||(u_0-\overline{u}_0)(\xi )-(u_0-\overline{u}_0)(x)| \,d\xi \,ds \nonumber \\&\quad \le \int _{t'}^t\int _\mathbb {R}K\left( \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}+ \Vert v-\overline{v}\Vert ^\beta _{1,\frac{1}{2}}\right) \Vert u_0\Vert _1|x-\xi | \nonumber \\&\quad \quad \left( \frac{1}{|x-\xi |^{\gamma (1-\beta )}s^{\frac{2-\gamma (1-\beta )}{2}}}+\frac{1}{s^{\frac{3}{2}}}\right) \text{ e }^{-C\frac{(x-\xi )^2}{s}}\,d\xi \,ds \nonumber \\&\quad \quad +\int _{t'}^t\int _\mathbb {R}\frac{K\Vert u_0-\overline{u}_0\Vert _1|x-\xi |}{s^{\frac{3}{2}}}\text{ e }^{-C\frac{(x-\xi )^2}{s}}\,d\xi \,ds \nonumber \\&\quad \le \int _{t'}^t\int _\mathbb {R}K\left( \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}+ \Vert v-\overline{v}\Vert ^\beta _{1,\frac{1}{2}}\right) \Vert u_0\Vert _1\left( \frac{T^{\frac{1}{2}}}{s}+\frac{1}{s}\right) \text{ e }^{-C\frac{(x-\xi )^2}{s}}\,d\xi \,ds \nonumber \\&\quad \quad +\int _{t'}^t\int _\mathbb {R}\frac{K\Vert u_0-\overline{u}_0\Vert _1}{s}\text{ e }^{-C\frac{(x-\xi )^2}{s}}\,d\xi \,ds\nonumber \\&\quad \le K\left( \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}+\Vert v-\overline{v}\Vert ^\beta _{1,\frac{1}{2}}+ \Vert u_0-\overline{u}_0\Vert _1\right) \int _{t'}^t\int _\mathbb {R}\frac{1}{s}\text{ e }^{-C\frac{(x-\xi )^2}{s}}\,d\xi \,ds \nonumber \\&\quad \le K\left( \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}+\Vert v-\overline{v}\Vert ^\beta _{1,\frac{1}{2}}+ \Vert u_0-\overline{u}_0\Vert _1\right) \int _{t'}^t\frac{1}{s^{\frac{1}{2}}}\,ds \nonumber \\&\quad \le K\left( \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}+\Vert v-\overline{v}\Vert ^\beta _{1,\frac{1}{2}}+ \Vert u_0-\overline{u}_0\Vert _1\right) (t-t')^{\frac{1}{2}}, \end{aligned}$$

(51)

where $K=K(R,\lambda ,T,\Vert u_0\Vert _1)$. From estimates (49), (50), and (51), we have

$$\begin{aligned} \Vert V\Vert _{1,\frac{1}{2}}\le K(R,\lambda ,T,\Vert u_0\Vert _1)\left( \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}+ \Vert v-\overline{v}\Vert ^\beta _{1,\frac{1}{2}}+\Vert u_0-\overline{u}_0\Vert _1\right) , \end{aligned}$$

(52)

with a new K. Similarly, we can estimate W:

$$\begin{aligned} W(x,t)&=\int _0^t\int _\mathbb {R}{\varGamma }_{[v]}(x,t,\xi ,\tau )f(\xi ,\tau )- {\varGamma }_{[\overline{v}]}(x,t,\xi ,\tau )\overline{f}(\xi ,\tau ) \,d\xi \,d\tau \\&=\int _0^t\int _\mathbb {R}({\varGamma }_{[v]}-{\varGamma }_{[\overline{v}]})(x,t,\xi ,\tau )f(\xi ,\tau )\\&\quad +\, {\varGamma }_{[\overline{v}]}(x,t,\xi ,\tau )(f-\overline{f})(\xi ,\tau ) \,d\xi \,d\tau . \end{aligned}$$

Hence, using Lemma 10 and (41), we have

$$\begin{aligned} |W(x,t)|&\le \int _0^t\int _\mathbb {R}|({\varGamma }_{[v]}-{\varGamma }_{[\overline{v}]})(x,t,\xi ,\tau )f(\xi ,\tau )|\, d\xi \, d\tau \nonumber \\&\quad +\int _0^t\int _\mathbb {R}|{\varGamma }_{[\overline{v}]}(x,t,\xi ,\tau )(f-\overline{f})(\xi ,\tau )| \,d\xi \,d\tau \nonumber \\&\le \int _0^t\int _\mathbb {R}K(\Vert v-\overline{v}\Vert _{1,\frac{1}{2}}\Vert f\Vert _\infty + \Vert f-\overline{f}\Vert _\infty )\frac{1}{(t-\tau )^{\frac{1}{2}}}\text{ e }^{-C\frac{(x-\xi )^2}{t-\tau }}\,d\xi \,d\tau \nonumber \\&\le K(R,\lambda ,T)(\Vert f\Vert _\infty +1)T\left( \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}+\Vert f-\overline{f}\Vert _\infty \right) . \end{aligned}$$

(53)

In addition,

$$\begin{aligned} |\partial _xW(x,t)|&\le \int _0^t\int _\mathbb {R}|\left( \partial _x{\varGamma }_{[v]}-\partial _x{\varGamma }_{[\overline{v}]}\right) (x,t,\xi ,\tau )f(\xi ,\tau )| \nonumber \\&\quad +\int _0^t\int _\mathbb {R}|\partial _x{\varGamma }_{[\overline{v}]}(x,t,\xi ,\tau )(f-\overline{f})(\xi ,\tau )| \,d\xi \,d\tau \nonumber \\&\le \int _0^t\int _\mathbb {R}K\left( \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}\Vert f\Vert _\infty + \Vert f-\overline{f}\Vert _\infty \right) \frac{1}{(t-\tau )}\text{ e }^{-C\frac{(x-\xi )^2}{t-\tau }}\,d\xi \,d\tau \nonumber \\&\le K(R,\lambda ,T)(\Vert f\Vert _\infty +1)T^{\frac{1}{2}}\left( \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}+\Vert f-\overline{f}\Vert _\infty \right) . \end{aligned}$$

(54)

To prove the Hölder continuity with respect to t, we write

$$\begin{aligned} W(x,t)-W(x,t')&=\int _{0}^t\int _\mathbb {R}[({\varGamma }_{[v]}(x,t,\xi ,\tau )]f(\xi ,\tau )\,d\xi \,d\tau \ \ \ \ \ \ \ \ \ \ \\&\quad -\int _{0}^t\int _\mathbb {R}{\varGamma }_{[\overline{v}]}(x,t,\xi ,\tau )\overline{f}(\xi ,\tau ) \,d\xi \,d\tau \\&\quad -\int ^{t'}_0\int _\mathbb {R}[({\varGamma }_{[v]}(x,t',\xi ,\tau )]f(\xi ,\tau ) \\&\quad - {\varGamma }_{[\overline{v}]}(x,t,\xi ',\tau )\overline{f}(\xi ,\tau ) \,d\xi \,d\tau \\&=\int _{t'}^t\int _\mathbb {R}({\varGamma }_{[v]}(x,t,\xi ,\tau )]f(\xi ,\tau ) \\&\quad - {\varGamma }_{[\overline{v}]}(x,t,\xi ,\tau )\overline{f}(\xi ,\tau )) \,d\xi \,d\tau \\&\quad +\int _0^{t'}\int _\mathbb {R}({\varGamma }_{[v]}(x,t,\xi ,\tau )- {\varGamma }_{[v]}(x,t',\xi ,\tau ))f(\xi ,\tau )\,d\xi \,d\tau \\&\quad -\int _0^{t'}\int _\mathbb {R}({\varGamma }_{[\overline{v}]}(x,t,\xi ,\tau )-{\varGamma }_{[\overline{v}]}(x,t',\xi ,\tau ))\overline{f}(\xi ,\tau ) \,d\xi \,d\tau \\&= W_1+W_2+W_3 \end{aligned}$$

where, for and $0<\epsilon <t'$ arbitrary, we set

$$\begin{aligned} W_1=&\int _{t'}^t\int _\mathbb {R}(({\varGamma }_{[v]}- {\varGamma }_{[\overline{v}]})(x,t,\xi ,\tau )]f(\xi ,\tau )+ {\varGamma }_{[\overline{v}]}(x,t,\xi ,\tau )(f-\overline{f})(\xi ,\tau )) \,d\xi \,d\tau \\ W_2=&\int _{t'-\epsilon }^{t'}\int _\mathbb {R}\left( {\varGamma }_{[v]}(x,t,\xi ,\tau )-{\varGamma }_{[v]}(x,t',\xi ,\tau )\right) f(\xi ,\tau ) \,d\xi \,d\tau \\&-\int _{t'-\epsilon }^{t'}\int _\mathbb {R}\left( {\varGamma }_{[\overline{v}]}(x,t,\xi ,\tau )-{\varGamma }_{[\overline{v}]}(x,t',\xi ,\tau )\right) \overline{f}(\xi ,\tau )] \,d\xi \,d\tau \end{aligned}$$

and

$$\begin{aligned} W_3=&\int _0^{t'-\epsilon }\int _\mathbb {R}\int _{t'}^t\left[ \partial _t{\varGamma }_{[v]}(x,\xi ,s,\tau )f(\xi ,\tau ) \,ds\,d\xi \,d\tau \right. \\&\left. -\int _0^{t'-\epsilon }\int _\mathbb {R}\partial _t{\varGamma }_{[\overline{v}]}(x,\xi ,s,\tau )\overline{f}(\xi ,\tau )\right] \,ds\,d\xi \,d\tau \, . \end{aligned}$$

Using Lemma 10, we estimate

$$\begin{aligned} |W_1|&\le \int _{t'}^t\int _\mathbb {R}(K\Vert v-\overline{v}\Vert _{1,\frac{1}{2}}\Vert f\Vert _\infty +K\Vert f- \overline{f}\Vert _\infty )\frac{\text{ e }^{-C\frac{(x-\xi )^2}{t-\tau }}}{(t-\tau )^{\frac{1}{2}}}\,d\xi \,d\tau \nonumber \\&\le K(R,\lambda ,T)(\Vert f\Vert _\infty +1)T^{\frac{1}{2}}\left( \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}+\Vert f- \overline{f}\Vert _\infty \right) (t-t')^{\frac{1}{2}}, \end{aligned}$$

(55)

and

$$\begin{aligned} |W_2|&\le \int _{t'-\epsilon }^{t'}\int _\mathbb {R}\left( \frac{K}{(t-\tau )^{\frac{1}{2}}}\text{ e }^{-C\frac{(x-\xi )^2}{t-\tau }}+ \frac{K}{(t'-\tau )^{\frac{1}{2}}}\text{ e }^ {-C\frac{(x-\xi )^2}{t'-\tau }}\right) \nonumber \\&\quad (\Vert f\Vert _\infty +\Vert \overline{f}\Vert _\infty ) \,d\xi \,d\tau \nonumber \\&\le K(\Vert f\Vert _\infty +\Vert \overline{f}\Vert _\infty )\epsilon . \end{aligned}$$

(56)

The term $W_3$ can be estimated as follows:

$$\begin{aligned} W_3&=\int _0^{t'-\epsilon }\int _\mathbb {R}\int _{t'}^t\left[ \partial _t{\varGamma }_{[v]}(x,\xi ,s,\tau )f(\xi ,\tau )- \partial _t{\varGamma }_{[\overline{v}]}(x,\xi ,s,\tau )\overline{f}(\xi ,\tau )\right] \,ds\,d\xi \,d\tau \\&=\int _0^{t'-\epsilon }\int _{t'}^t\int _\mathbb {R}\left[ (\partial _t{\varGamma }_{[v]}- \partial _t{\varGamma }_{[\overline{v}]})(x,\xi ,s,\tau )(f(\xi ,\tau )-f(x,\tau ))\right. \\&\quad \left. + \,\partial _t{\varGamma }_{[\overline{v}]}(x,\xi ,s,\tau )((f-\overline{f})(\xi ,\tau )- (f-\overline{f})(x,\tau ))\right] \,d\xi \,ds \,d\tau . \end{aligned}$$

Now, applying Lemma 10 again, and writing $K_1=K(\Vert v-\overline{v}\Vert _{1,\frac{1}{2}}+\Vert v-\overline{v}\Vert _{1,\frac{1}{2}}^\beta )\Vert f\Vert _{1,\frac{1}{2}}$, it follows that $|W_3|$

$$\begin{aligned}&\le \int _0^{t'-\epsilon }\int _{t'}^t\int _\mathbb {R}K_1\left( \frac{1}{|x-\xi |^{\gamma (1-\beta )}(s-\tau )^ {\frac{2-\gamma (1-\beta )}{2}}}+ \frac{1}{(s-\tau )^{\frac{3}{2}}}\right) \text{ e }^{-C\frac{(x-\xi )^2}{s-\tau }}|x-\xi | \nonumber \\&\quad +\frac{K}{(s-\tau )^{\frac{3}{2}}}\text{ e }^{-C\frac{(x-\xi )^2}{s-\tau }} \Vert f-\overline{f}\Vert _{1,\frac{1}{2}}|x-\xi |\,d\xi \,ds \,d\tau \nonumber \\&\le K\left( 1+\Vert f\Vert _{1,\frac{1}{2}}\right) \left[ \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}+\Vert v-\overline{v}\Vert _{1,\frac{1}{2}}^\beta \right. \nonumber \\&\quad \left. +\Vert f-\overline{f}\Vert _{1,\frac{1}{2}}\right] \int _0^{t'-\epsilon }\int _{t'}^t\int _\mathbb {R}\left( \frac{|x-\xi |^ {1-\gamma (1-\beta )}}{(s-\tau )^{\frac{2-\gamma (1-\beta )}{2}}}+\frac{|x-\xi |}{(s-\tau )^{\frac{3}{2}}}\right) \text{ e }^ {-C\frac{(x-\xi )^2}{s-\tau }}\,d\xi \,ds \,d\tau \nonumber \\&\le K\left( 1+\Vert f\Vert _{1,\frac{1}{2}}\right) \left[ \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}+\Vert v-\overline{v}\Vert _{1,\frac{1}{2}}^\beta \right. \nonumber \\&\quad \left. +\Vert f-\overline{f}\Vert _{1,\frac{1}{2}}\right] \int _0^{t'-\epsilon }\int _{t'}^t\int _\mathbb {R}\left( \frac{1}{(s-\tau )^{\frac{1}{2}}}+ \frac{1}{s-\tau }\right) \text{ e }^{-C\frac{(x-\xi )^2}{s-\tau }}\,d\xi \,ds \,d\tau \nonumber \\&\le K\left( 1+\Vert f\Vert _{1,\frac{1}{2}}\right) \left[ \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}+\Vert v-\overline{v}\Vert _{1,\frac{1}{2}}^\beta \right. \nonumber \\&\quad \left. +\Vert f-\overline{f}\Vert _{1,\frac{1}{2}}\right] \left( T^{\frac{1}{2}}+1\right) \int _0^{t'-\epsilon }\int _{t'}^t\int _\mathbb {R}\frac{1}{s-\tau }\text{ e }^ {-C\frac{(x-\xi )^2}{s-\tau }}\,d\xi \,ds \,d\tau \nonumber \\&\le K(1+\Vert f\Vert _{1,\frac{1}{2}})\left[ \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}+\Vert v-\overline{v}\Vert _{1,\frac{1}{2}}^ \beta +\Vert f-\overline{f}\Vert _{1,\frac{1}{2}}\right] \, \cdot \nonumber \\&\quad \cdot \, \int _0^{t'-\epsilon }\int _{t'}^t \frac{1}{(s-\tau )^{\frac{1}{2}}} \,ds \,d\tau \nonumber \\&\le K\left( 1+\Vert f\Vert _{1,\frac{1}{2}}\right) \left[ \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}+\Vert v-\overline{v}\Vert _{1,\frac{1}{2}}^\beta + \Vert f-\overline{f}\Vert _{1,\frac{1}{2}}\right] T(t-t')^{\frac{1}{2}}, \end{aligned}$$

(57)

where for the last inequality we used that (56) is true for all $\epsilon \in (0,t')$. From (55), (56), and (57), we conclude that

$$\begin{aligned}&|W(x,t)-W(x,t')| \nonumber \\&\quad \le K\left( \Vert f\Vert _{1,\frac{1}{2}}+1\right) \left( \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}+\Vert v-\overline{v}\Vert _{1,\frac{1}{2}}^\beta + \Vert f-\overline{f}\Vert _{1,\frac{1}{2}}\right) \nonumber \\&\qquad T^{\frac{1}{2}}(t-t')^{\frac{1}{2}}, \end{aligned}$$

(58)

where $K=K(R,\lambda ,T)$. It follows from (53), (54), and (58) that

$$\begin{aligned} \Vert W\Vert _{1,\frac{1}{2}}\le K(R,\lambda ,T)T^{\frac{1}{2}} \left( \Vert f\Vert _{1,\frac{1}{2}}+1\right) \left( \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}+ \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}^\beta +\Vert f-\overline{f}\Vert _{1,\frac{1}{2}}\right) .\nonumber \\ \end{aligned}$$

(59)

Finally, from (52) and (59), we have

$$\begin{aligned}&\Vert u-\overline{u}\Vert _{1,\frac{1}{2}}\le \ \Vert V\Vert _{1,\frac{1}{2}}+\Vert W\Vert _{1,\frac{1}{2}} \nonumber \\&\quad \le K\left( \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}+\Vert v-\overline{v}\Vert _{1,\frac{1}{2}}^\beta +\Vert u_0-\overline{u}_0\Vert _1 \right. \nonumber \\&\qquad \left. + T^{\frac{1}{2}}\left( \Vert f\Vert _{1,\frac{1}{2}}+1\right) \left( \Vert f-\overline{f}\Vert _{1,\frac{1}{2}}+\Vert v-\overline{v}\Vert _{1,\frac{1}{2}}+ \Vert v-\overline{v}\Vert _{1,\frac{1}{2}}^\beta \right) \right) , \end{aligned}$$

(60)

where $K=K(R,\lambda ,T,\Vert u_0\Vert _1)$. $\square$