Consider the initial-boundary value problem
$$\begin{aligned}& v_{\varepsilon t} -\operatorname{div} \bigl( \bigl(a(x)+ \varepsilon \bigr) \bigl( \vert v_{ \varepsilon} \vert ^{\alpha (x)}+ \varepsilon \bigr) \vert {\nabla v_{ \varepsilon}} \vert ^{p(x)- 2}{ \nabla v_{\varepsilon }} \bigr)- \sum_{i=1}^{N}g^{i}(x,t,v_{\varepsilon}) \frac{\partial v_{\varepsilon}}{\partial x_{i}} \\& \quad =d(x,t,v_{\varepsilon}) ,\quad (x,t)\in {Q_{T}}, \end{aligned}$$
(23)
$$\begin{aligned}& {v_{\varepsilon}}(x,t) = 0,\quad (x,t) \in \partial \Omega \times (0,T), \end{aligned}$$
(24)
$$\begin{aligned}& {v_{\varepsilon}}(x,0) = {v_{\varepsilon 0}}(x),\quad x \in \Omega , \end{aligned}$$
(25)
where \(v_{\varepsilon 0} \in C^{\infty}_{0}(\Omega )\), \(\|v_{\varepsilon 0}\|_{L^{\infty}(\Omega )}\leq \|v_{0}\|_{L^{\infty}( \Omega )}\), \(a(x) \vert \nabla v_{\varepsilon 0} \vert ^{p(x)}\) is uniformly convergent to \(a(x)|\nabla v_{0}(x)|^{p(x)}\) in \({L^{1}}(\Omega )\).
Definition 5
A function \(v(x,t)\in \mathbf{W}(Q_{T})\bigcap L^{\infty}(0,T;L^{2}(\Omega ))\) is said to be a weak solution of problem (23)–(25) if
$$ \frac{\partial v}{\partial t}\in \mathbf{W}'(Q_{T}),\ $$
and for any function \(\varphi \in C_{0}^{1}({Q_{T}})\),
$$ \begin{aligned}[b] & \iint _{{Q_{T}}} \biggl(\frac{\partial v}{\partial t} \varphi +\bigl(a(x)+ \varepsilon \bigr) \bigl( \vert v \vert ^{\alpha (x)}+\varepsilon \bigr) \vert \nabla v \vert ^{p(x)- 2}\nabla v \nabla \varphi \biggr)\,dx\,dt \\ &\quad = \iint _{{Q_{T}}} \Biggl[\sum_{i=1}^{N}g^{i}(x,t,v) \frac{\partial v}{\partial x_{i}}+d(x,t,v) \Biggr] \varphi \,dx\,dt . \end{aligned} $$
(26)
The initial value (24) is satisfied in the sense as (12).
Then, by a similar method as that in [3], we have the following theorem.
Theorem 6
If \(a(x)\in C^{1}(\overline{\Omega})\) satisfies (6), \(g^{i}(x,t,s)\) and \(d(x,t,s)\) satisfy (13)–(14), there is a weak solution \(v_{\varepsilon}\) of the initial boundary value problem (23)–(24) on \(\Omega \times [0, T^{*})\), where
$$ T^{*}=\sup \bigl\{ \theta : \Vert u \Vert _{\infty ,Q_{\theta}}< \infty \bigr\} . $$
(27)
Firstly, we quote the following lemmas.
Lemma 7
If \(u_{\varepsilon}\in L^{\infty}(0,T;L^{2}(\Omega ))\bigcap \mathbf{W}(Q_{T})\), \(\| u_{\varepsilon t}\|_{\mathbf{W}'(Q_{T})}\leq c\), \(\|\nabla (|u_{\varepsilon}|^{r-1}u_{\varepsilon})\|_{p,Q_{T}}\leq c\), then there is a subsequence of \(\{u_{\varepsilon}\}\) which is relatively compact in \(L^{s}(Q_{T})\) with \(s\in (1,\infty )\). Here, \(r\geq 1\), \(p>1\).
This lemma comes from [19, Sect. 8].
Lemma 8
Suppose that \(p(x)\in C(\overline{\Omega})\) is local Hölder continuous, and denote that
$$ p^{+}=\max_{x\in \overline{\Omega}}p(x),\qquad p^{-}=\min _{x\in \overline{\Omega}}p(x). $$
Then the following facts are true.
(i) The space \((L^{p(x)}(\Omega ), \|\cdot \|_{L^{p(x)}(\Omega )} )\), \((W^{1,p(x)}(\Omega ), \|\cdot \|_{W^{1,p(x)}(\Omega )} )\) and \(W^{1,p(x)}_{0}(\Omega )\) are reflexive Banach spaces.
(ii) \(p(x)\)-Hölder’s inequality. Let \(q(x)=\frac{p(x)}{p(x)-1}\). Then the conjugate space of \(L^{p(x)}(\Omega )\) is \(L^{q(x)}(\Omega )\). For any \(u \in L^{p(x)}(\Omega )\) and \(v \in L^{q(x)}(\Omega )\), there holds
$$ \biggl\vert \int _{\Omega}uv \,dx \biggr\vert \leq 2 \Vert u \Vert _{L^{p(x)}(\Omega )} \Vert v \Vert _{L^{q(x)}(\Omega )}. $$
(iii)
If \(\Vert u \Vert _{L^{p(x)}(\Omega )} = 1\), then \(\int _{ \Omega} \vert u \vert ^{p(x)} \,dx = 1\).
If \(\Vert u \Vert _{L^{p(x)}(\Omega )} > 1\), then \(\Vert u \Vert ^{p^{-}}_{L^{p(x)}( \Omega )}\leq \int _{\Omega} \vert u \vert ^{p(x)} \,dx\leq \Vert u \Vert ^{p^{+}}_{L^{p(x)}( \Omega )}\).
If \(\Vert u \Vert _{L^{p(x)}(\Omega )} < 1\), then \(\Vert u \Vert ^{p^{+}}_{L^{p(x)}( \Omega )}\leq \int _{\Omega} \vert u \vert ^{p(x)} \,dx\leq \Vert u \Vert ^{p^{-}}_{L^{p(x)}( \Omega )}\).
This lemma can be found in [9, 20] etc.
Secondly, we give the details of the proof of Theorem 2.
Proof of Theorem 2
According to Theorem 6, there is a weak solution \(v_{\varepsilon}\) of the initial boundary value problem (23)–(24), and
$$ \Vert v_{\varepsilon} \Vert _{\infty , Q_{T_{0}}}\leq c(T_{0}), $$
where \(T_{0}< T^{*}\) is a given positive constant, \(c(T_{0})\) is a constant that may depend on \(T_{0}\).
By multiplying (23) by \(v_{\varepsilon}\), one has
$$ \begin{aligned}[b] &\frac{1}{2} \int _{\Omega}v_{\varepsilon}^{2}\,dx + \int _{0}^{T_{0}} \int _{\Omega}\bigl(a(x)+\varepsilon \bigr) \bigl( \vert v_{\varepsilon} \vert ^{\alpha (x)}+ \varepsilon \bigr) \vert \nabla v_{\varepsilon} \vert ^{p(x)}\,dx\,dt \\ &\quad =\frac{1}{2} \int _{\Omega}v_{\varepsilon 0}^{2}\,dx+ \iint _{Q_{T_{0}}}d(x,t,v_{ \varepsilon})v_{\varepsilon}\,dx\,dt +\sum_{i=i}^{N} \int _{0}^{T_{0}} \int _{\Omega}g^{i}(x,t, v_{\varepsilon})v_{\varepsilon}\frac{\partial v_{\varepsilon}}{\partial x_{i}} \,dx\,dt. \end{aligned} $$
(28)
Since
$$ \bigl\vert g^{i}(x,t, v_{\varepsilon}) \bigr\vert \leq g(x,t) \vert v_{\varepsilon} \vert ^{ \frac{\alpha (x)}{p(x)}},\quad i=1,2,\ldots , N, $$
and \(g(x,t)\in C(\overline{Q_{T_{0}}})\) satisfies (13), then one has
$$ \begin{aligned}[b] \biggl\vert g^{i}(x,t, v_{\varepsilon})v_{\varepsilon}\frac{\partial v_{\varepsilon}}{\partial x_{i}} \biggr\vert \leq{} & \biggl\vert g(x,t) \vert v_{\varepsilon} \vert ^{\frac{\alpha (x)}{p(x)}} \frac{\partial v_{\varepsilon}}{\partial x_{i}} \biggr\vert \leq c(\varepsilon )+\varepsilon \vert v \vert ^{\alpha (x)} \vert \nabla v_{ \varepsilon} \vert ^{p(x)}. \end{aligned} $$
(29)
By that \(|d(x,t,s)|\leq d_{0}|s|^{\sigma -1}+h(x,t)\), \(\|h\|_{L^{1}(0,\theta ;L^{\infty}(\Omega ))}\leq c\), one has
$$ \begin{aligned}[b] \biggl\vert \iint _{{Q_{T_{0}}}}d(x,t,v_{\varepsilon})v_{\varepsilon}\,dx\,dt \biggr\vert & \leq \iint _{{Q_{T_{0}}}} \bigl[d_{0} \vert s \vert ^{\sigma -1}+h(x,t) \bigr] \vert v_{\varepsilon} \vert \,dx\,dt \\ & \leq c(T_{0}) \iint _{{Q_{T_{0}}}} \bigl[d_{0} \vert s \vert ^{\sigma -1}+h(x,t) \bigr]\,dx\,dt \\ & \leq c(T_{0}). \end{aligned} $$
(30)
Then formulas (28),(29), and (30) imply
$$ \begin{aligned}[b] \iint _{{Q_{T_{0}}}} a(x) \vert v_{\varepsilon} \vert ^{\alpha (x)} \vert \nabla v_{ \varepsilon} \vert ^{p(x)}\,dx\,dt &\leq \iint _{{Q_{T_{0}}}} \bigl(a(x)+\varepsilon \bigr) \bigl( \vert v_{ \varepsilon} \vert ^{\alpha (x)}+\varepsilon \bigr) \vert \nabla v_{\varepsilon} \vert ^{p(x)}\,dx\,dt \\ &\leq c(T_{0}), \end{aligned} $$
(31)
accordingly, one has
$$ \begin{aligned}[b] \iint _{Q_{T_{0}}}a(x) \bigl\vert \nabla v_{\varepsilon}^{ \frac{\alpha (x)}{p(x)}+1} \bigr\vert ^{p(x)}\,dx\,dt \leq{}& c \iint _{Q_{T_{0}}}a(x) \biggl[ \biggl\vert \nabla \frac{\alpha (x)}{p(x)} \biggr\vert \bigl\vert v_{\varepsilon}^{ \frac{\alpha (x)}{p(x)}+1}\ln v_{\varepsilon}(x) \bigr\vert \\ & {}+ \biggl(\frac{\alpha (x)}{p(x)}+1 \biggr)v_{\varepsilon}^{ \frac{\alpha (x)}{p(x)}} \vert \nabla v_{\varepsilon} \vert \biggr]^{p(x)}\,dx\,dt \\ \leq{}& c+c \iint _{Q_{T_{0}}} a(x) \vert v_{\varepsilon} \vert ^{\alpha (x)} \vert \nabla v_{\varepsilon} \vert ^{p(x)}\,dx\,dt \\ \leq{}& c(T_{0}). \end{aligned} $$
(32)
Now, for any \(u\in C_{0}^{1}(Q_{T_{0}})\), \(\|u\|_{W(Q_{T_{0}})}=1\), one has
$$ \begin{aligned}[b] \langle v_{\varepsilon t}, u\rangle ={}&{-} \iint _{Q_{T_{0}}}\bigl(a(x)+\varepsilon \bigr) \bigl( \vert v_{\varepsilon} \vert ^{\alpha (x)}+ \varepsilon \bigr) \vert {\nabla v_{\varepsilon}} \vert ^{p(x)- 2}{ \nabla v_{\varepsilon }}\nabla u \,dx\,dt \\ &{}+\sum_{i=1}^{N} \iint _{Q_{T_{0}}}g^{i}(x,t,v_{\varepsilon}) \frac{\partial v_{\varepsilon}}{\partial x_{i}}u\,dx\,dt+ \iint _{Q_{T_{0}}}d(x,t,v_{ \varepsilon})u\,dx\,dt. \end{aligned} $$
(33)
Since \(g^{i}(x,t, v_{\varepsilon})\) and \(d(x,t,v_{\varepsilon})\) satisfy (13)(14), by \(\|v_{\varepsilon}\|_{\infty , Q_{T_{0}}}\leq c(T_{0})\), using the Hölder inequality, one obtains
$$ \begin{aligned} \biggl\vert \iint _{Q_{T_{0}}}g^{i}(x,t,v_{\varepsilon}) \frac{\partial v_{\varepsilon}}{\partial x_{i}}u\,dx\,dt \biggr\vert &\leq \iint _{Q_{T_{0}}}g(x,t) \vert v_{\varepsilon} \vert ^{ \frac{\alpha (x)}{p(x)}-1} \biggl\vert \frac{\partial v_{\varepsilon}}{\partial x_{i}}u \biggr\vert \,dx\,dt \\ &\leq c(T_{0}) \biggl( \iint _{Q_{T_{0}}} \biggl( \frac{g(x,t)^{p(x)}}{a(x)} \biggr)^{\frac{1}{p(x)-1}}\,dx\,dt \biggr)^{ \frac{1}{q_{1}}} \\ &\leq c(T_{0}), \end{aligned} $$
where \(q_{1}=\max_{x\in \overline{\Omega}}\frac{p(x)}{p(x)-1}\) or \(\min_{x\in \overline{\Omega}}\frac{p(x)}{p(x)-1}\) according to (iii) of Lemma 8, and
$$ \begin{aligned} \biggl\vert \iint _{Q_{T_{0}}}d(x,t,v_{\varepsilon})u\,dx\,dt \biggr\vert &\leq \iint _{Q_{T_{0}}} \bigl(d_{0} \vert v_{\varepsilon} \vert ^{\sigma -1}+h(x,t) \bigr) \vert u \vert \,dx\,dt \leq c(T_{0}). \end{aligned} $$
By the above discussion, one has
$$ \begin{aligned} \bigl\vert \langle v_{\varepsilon t}, u\rangle \bigr\vert \leq{} &c(T_{0}) \biggl[ \iint _{{Q_{T_{0}}}}\bigl(a(x)+\varepsilon \bigr) \bigl( \vert v_{ \varepsilon} \vert ^{\alpha (x)}+\varepsilon \bigr) \vert \nabla v_{\varepsilon} \vert ^{p(x)}\,dx\,dt\\ &{}+ \iint _{{Q_{T_{0}}}} \bigl( \vert u \vert ^{p(x)}+ \vert \nabla u \vert ^{p(x)} \bigr)\,dx\,dt+1 \biggr] \\ \leq {}&c(T_{0}). \end{aligned} $$
Since \(C_{0}^{1}(Q_{T_{0}})\) is dense on \(W(Q_{T_{0}})\), one has
$$ \Vert v_{\varepsilon t} \Vert _{\mathbf{W}'(Q_{T_{0}})}\leq c(T_{0}) $$
and
$$ \bigl\Vert v^{\frac{\alpha (x)}{p(x)}+1}_{\varepsilon t} \bigr\Vert _{ \mathbf{W}'(Q_{T_{0}})}\leq c(T_{0}). $$
(34)
If one denotes \(d(x)=\text{dist}(x,\partial \Omega )\) as the distance function from the boundary ∂Ω, sets
$$ \Omega _{\lambda}=\bigl\{ x\in \Omega : d(x)>\lambda \bigr\} $$
for small \(\lambda >0\), and defines \(\varphi \in C_{0}^{1}(\Omega )\), \(0\leq \varphi \leq 1\) such that
$$ \varphi \mid _{\Omega _{2\lambda}}=1, \qquad \varphi \mid _{\Omega \setminus \Omega _{\lambda}}=0, $$
(35)
then, for any \(\varphi \in C_{0}^{1}(\Omega )\) satisfying (35), \(0\leq \varphi \leq 1\), one has
$$ \bigl\Vert \bigl(\varphi v_{\varepsilon}^{\frac{\alpha (x)}{p(x)}+1} \bigr)_{t} \bigr\Vert _{\mathbf{W}'(Q_{T_{0}})}\leq c( \lambda ,T_{0}). $$
(36)
Once again, since \(a(x)>0\) when \(x\in \Omega \), by (32), one has
$$ \iint _{Q_{T_{0}}} \bigl\vert \nabla \bigl(\varphi v_{\varepsilon}^{ \frac{\alpha (x)}{p(x)}+1} \bigr) \bigr\vert ^{p(x)}\,dx\,dt\leq c(\lambda ,T_{0}) \biggl(1+ \int _{0}^{T} \int _{D_{\lambda}} \vert \nabla v_{\varepsilon} \vert ^{p(x)} \,dx\,dt\biggr) \leq c(\lambda ,T_{0}), $$
(37)
and so
$$ \bigl\Vert \nabla \bigl(\varphi v_{\varepsilon}^{ \frac{\alpha (x)}{p(x)}+1} \bigr) \bigr\Vert _{p^{-},Q_{T_{0}}}\leq \bigl\Vert \nabla \bigl(\varphi v_{\varepsilon}^{ \frac{\alpha (x)}{p(x)}+1} \bigr) \bigr\Vert _{p(x),Q_{T_{0}}}\leq c(T_{0}). $$
(38)
If one denotes \(v_{1\varepsilon}=v_{\varepsilon}^{\frac{\alpha (x)}{p(x)}+1}\), then, from (36) and (38), Lemma 7 yields that \(\varphi v_{1\varepsilon}\rightarrow \varphi v_{1}\) a.e. in \(Q_{T}\). By the arbitrariness of φ, one has \(v_{1\varepsilon}=v_{\varepsilon}^{\frac{\alpha (x)}{p(x)}+1} \rightarrow v_{1}\) a.e. in \(Q_{T_{0}}\). By (12), \(v \in L^{\infty}(Q_{T_{0}})\) and
$$ v_{\varepsilon} \rightharpoonup v, \quad \text{weakly star in } L^{\infty}(Q_{T_{0}}). $$
(39)
By the weak convergence theory, one has
$$ v_{1}=v^{\frac{\alpha (x)}{p(x)}+1}. $$
Thus, \(v_{\varepsilon}\rightarrow v\) a.e. in \(Q_{T_{0}}\), and then
$$ g^{i}(x,t, v_{\varepsilon})\rightarrow g^{i}(x,t,v),\qquad d(x,t, v_{ \varepsilon})\rightarrow d(x,t,v),\quad \text{a.e. in } Q_{T_{0}}. $$
(40)
Moreover, since \(a(x)\in C^{1}(\overline{\Omega})\) and \(a(x)|_{x\in \Omega}>0\), one has
$$ \nabla v_{\varepsilon}^{\frac{\alpha (x)}{p(x)}+1}\rightharpoonup \nabla v^{\frac{\alpha (x)}{p(x)}+1}\quad \text{in } L^{1} \bigl(0,T; L^{p(x)}_{\mathrm{loc}}( \Omega ) \bigr). $$
(41)
Now, similar as the techniques used in [22, 27, 28, 30], if one chooses \((v_{\varepsilon}^{\frac{\alpha (x)}{p(x)}+1}-v^{ \frac{\alpha (x)}{p(x)}+1} )\phi \) as the test function where \(\phi (x)\in C_{0}^{1}(\Omega )\), then there holds
$$\begin{aligned}& \int _{0}^{{T_{0}}} \int _{{\Omega}} \frac{\partial v_{\varepsilon}}{\partial t} \bigl(v_{\varepsilon}^{ \frac{\alpha (x)}{p(x)}+1}-v^{\frac{\alpha (x)}{p(x)}+1} \bigr)\phi \,dx\,dt \\& \qquad {}+ \int _{0}^{{T_{0}}} \int _{{\Omega}}\phi (x) \bigl(a(x)+\varepsilon \bigr) \bigl( \vert v_{ \varepsilon} \vert ^{\alpha (x)}+\varepsilon \bigr) \vert \nabla v_{\varepsilon} \vert ^{p(x)-2} \nabla v_{\varepsilon}\nabla \bigl(v_{\varepsilon}^{ \frac{\alpha (x)}{p(x)}+1}-v^{\frac{\alpha (x)}{p(x)}+1} \bigr)\,dx\,dt \\& \qquad {}+ \int _{0}^{{T_{0}}} \int _{{\Omega}}\bigl(a(x)+\varepsilon \bigr) \bigl( \vert v_{ \varepsilon} \vert ^{\alpha (x)}+\varepsilon \bigr) \vert \nabla v_{\varepsilon} \vert ^{p(x)-2} \nabla v_{\varepsilon} \bigl(v_{\varepsilon}^{\frac{\alpha (x)}{p(x)}+1}-v^{ \frac{\alpha (x)}{p(x)}+1} \bigr)\nabla \phi \,dx\,dt \\& \qquad {}-\sum_{i=1}^{N} \int _{0}^{{T_{0}}} \int _{{\Omega}}g^{i}(x,t,v_{ \varepsilon}) \frac{\partial v_{\varepsilon}}{\partial x_{i}} \bigl(v_{ \varepsilon}^{\frac{\alpha (x)}{p(x)}+1}-v^{\frac{\alpha (x)}{p(x)}+1} \bigr)\phi \,dx\,dt \\& \quad = \int _{0}^{T_{0}} \int _{{\Omega}}d(x,t,v_{\varepsilon}) \bigl(v_{ \varepsilon}^{\frac{\alpha (x)}{{\alpha (x)}}+1}-v^{ \frac{\alpha (x)}{{\alpha (x)}}+1} \bigr)\phi \,dx\,dt. \end{aligned}$$
(42)
Since
$$ \bigl\vert g^{i}(x,t, v_{\varepsilon}) \bigr\vert \leq g(x,t) \vert v_{\varepsilon} \vert ^{ \frac{\alpha (x)}{p(x)}},\quad i=1,2,\ldots , N, $$
and using (14), it can be deduced that
$$ \int _{0}^{T_{0}} \int _{{\Omega}}\phi (x)a(x) \vert v_{\varepsilon} \vert ^{ \alpha _{(}x)} \vert \nabla v_{\varepsilon } \vert ^{p(x)} \nabla v_{\varepsilon} \nabla v^{\frac{\alpha (x)}{p(x)}+1}\,dx\,dt \leq c. $$
(43)
By the arbitrariness of ϕ and \(|v_{\varepsilon}|^{\alpha (x)} \vert \nabla v_{\varepsilon} \vert ^{p(x)-2} \nabla v_{\varepsilon}\in L^{1} (0,{T_{0}}; L_{\mathrm{loc}}^{ \frac{p(x)}{p(x)-1}}(\Omega ) )\), one has
$$ \nabla v\in L^{\infty}\bigl(0,T; L^{p(x)}_{\mathrm{loc}}( \Omega )\bigr). $$
(44)
By this property, one can show that
$$ g^{i}(x,t,v_{\varepsilon}) \frac{\partial v_{\varepsilon}}{\partial x_{i}} \rightharpoonup g^{i}(x,t,v) \frac{\partial v}{\partial x_{i}} \quad \text{in } L^{1}(Q_{T_{0}}). $$
(45)
The details are omitted here.
Thus, there are functions \(v(x,t)\) and \(\zeta _{i}\) satisfying
$$ v(x,t) \in L^{\infty}(Q_{T_{0}}),\quad \bigl\vert \zeta _{i}(x,t) \bigr\vert \in L^{1} \bigl(0,{T_{0}};L^{\frac{p(x)}{p(x) - 1}}( \Omega ) \bigr) $$
such that
$$\begin{aligned}& {v_{\varepsilon}} \rightharpoonup v, \quad \text{weakly star in } {L^{\infty}(Q_{T_{0}})}, \\& g^{i}(x,t,v_{\varepsilon})\rightarrow g^{i}(x,t, v),\qquad d(x,t,v_{ \varepsilon})\rightarrow d(x,t, v),\quad \text{a.e. in } Q_{T_{0}}, \\& \bigl(a(x)+\varepsilon \bigr) \bigl( \vert v_{\varepsilon} \vert ^{\alpha (x)}+\varepsilon \bigr) \vert {\nabla v_{\varepsilon}} \vert ^{p(x)- 2}{\nabla v_{ \varepsilon }}\rightharpoonup \vec{\zeta}, \quad \text{in } L^{1} \bigl(0,{T_{0}};L^{ \frac{p(x)}{p(x) - 1}}( \Omega ) \bigr). \end{aligned}$$
Moreover, by the important property (44), it is not difficult to show that
$$ \iint _{Q_{T_{0}}} a(x) \vert v \vert ^{\alpha (x)} \vert {\nabla v} \vert ^{p(x)- 2}{\nabla v}\cdot \nabla \varphi \,dx\,dt = \iint _{Q_{T_{0}}} \overrightarrow{\zeta} \cdot \nabla \varphi \,dx\,dt $$
for any given function \(\varphi \in C_{0}^{1} ({Q_{T_{0}}})\). Then v is a weak solution of equation (2) with the initial value (3). □