Existence of solution for a class of heat equation with double criticality
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We study the homogeneous Dirichlet problem for the equation $$\begin{aligned} u_t-{\text {div}}\left( \mathcal {F}(z,\nabla u)\nabla u\right) =f, \quad z=(x,t)\in Q_T=\Omega \times (0,T), \end{aligned}$$ u t - div F ( z , ∇ u ) ∇ u = f , z = ( x , t ) ∈ Q T = Ω × ( 0 , T ) , where $$\Omega \subset \mathbb {R}^N$$ Ω ⊂ R N , is a bounded domain with $$\partial \Omega \in C^2$$ ∂ Ω ∈ C 2 , and $$\mathcal {F}(z,\xi )=a(z)\vert \xi \vert ^{p(z)-2}+b(z)\vert \xi \vert ^{q(z)-2}$$ F ( z , ξ ) = a ( z ) | ξ | p ( z ) - 2 + b ( z ) | ξ | q ( z ) - 2 . The variable exponents p, q and the nonnegative modulating coefficients a, b are given Lipschitz-continuous functions. It is assumed that $$\frac{2N}{N+2}<p(z),\ q(z)$$ 2 N N + 2 < p ( z ) , q ( z ) , and that the modulating coefficients and growth exponents satisfy the balance conditions $$\begin{aligned} a(z)+b(z)\ge \alpha >0,\quad \vert p(z)-q(z)\vert <\frac{2}{N+2}\hbox { in }\overline{Q}_T \end{aligned}$$ a ( z ) + b ( z ) ≥ α > 0 , | p ( z ) - q ( z ) | < 2 N + 2 in Q ¯ T with $$\alpha =const$$ α = c o n s t . We find conditions on the source f and the initial data $$u(\cdot ,0)$$ u ( · , 0 ) that guarantee the existence of a unique strong solution u with $$u_t\in L^2(Q_T)$$ u t ∈ L 2 ( Q T ) and $$a\vert \nabla u\vert ^{p}+b\vert \nabla u\vert ^q\in L^\infty (0,T;L^1(\Omega ))$$ a | ∇ u | p + b | ∇ u | q ∈ L ∞ ( 0 , T ; L 1 ( Ω ) ) . The solution possesses the property of global higher integrability of the gradient, $$\begin{aligned} \vert \nabla u\vert ^{\min \{p(z),q(z)\}+r}\in L^1(Q_T)\quad \text {with any }r\in \left( 0,\frac{4}{N+2}\right) , \end{aligned}$$ | ∇ u | min { p ( z ) , q ( z ) } + r ∈ L 1 ( Q T ) with any r ∈ 0 , 4 N + 2 , which is derived with the help of new interpolation inequalities in the variable Sobolev spaces. The global second-order differentiability of the strong solution is proven: $$\begin{aligned} D_i\left( \sqrt{\mathcal {F}(z,\nabla u)}D_j u\right) \in L^{2}(Q_T),\quad i=1,2,\ldots ,N. \end{aligned}$$ D i F ( z , ∇ u ) D j u ∈ L 2 ( Q T ) , i = 1 , 2 , … , N . The same results are obtained for the equation with the regularized flux $$\mathcal {F}(z,\sqrt{\epsilon ^2+(\xi ,\xi )})\xi $$ F ( z , ϵ 2 + ( ξ , ξ ) ) ξ .
We study the homogeneous Dirichlet problem for the equation $$\begin{aligned} u_t-{\text {div}}\left( \mathcal {F}(z,\nabla u)\nabla u\right) =f, \quad z=(x,t)\in Q_T=\Omega \times (0,T), \end{aligned}$$ u t - div F ( z , ∇ u ) ∇ u = f , z = ( x , t ) ∈ Q T = Ω × ( 0 , T ) , where $$\Omega \subset \mathbb {R}^N$$ Ω ⊂ R N , is a bounded domain with $$\partial \Omega \in C^2$$ ∂ Ω ∈ C 2 , and $$\mathcal {F}(z,\xi )=a(z)\vert \xi \vert ^{p(z)-2}+b(z)\vert \xi \vert ^{q(z)-2}$$ F ( z , ξ ) = a ( z ) | ξ | p ( z ) - 2 + b ( z ) | ξ | q ( z ) - 2 . The variable exponents p, q and the nonnegative modulating coefficients a, b are given Lipschitz-continuous functions. It is assumed that $$\frac{2N}{N+2}<p(z),\ q(z)$$ 2 N N + 2 < p ( z ) , q ( z ) , and that the modulating coefficients and growth exponents satisfy the balance conditions $$\begin{aligned} a(z)+b(z)\ge \alpha >0,\quad \vert p(z)-q(z)\vert <\frac{2}{N+2}\hbox { in }\overline{Q}_T \end{aligned}$$ a ( z ) + b ( z ) ≥ α > 0 , | p ( z ) - q ( z ) | < 2 N + 2 in Q ¯ T with $$\alpha =const$$ α = c o n s t . We find conditions on the source f and the initial data $$u(\cdot ,0)$$ u ( · , 0 ) that guarantee the existence of a unique strong solution u with $$u_t\in L^2(Q_T)$$ u t ∈ L 2 ( Q T ) and $$a\vert \nabla u\vert ^{p}+b\vert \nabla u\vert ^q\in L^\infty (0,T;L^1(\Omega ))$$ a | ∇ u | p + b | ∇ u | q ∈ L ∞ ( 0 , T ; L 1 ( Ω ) ) . The solution possesses the property of global higher integrability of the gradient, $$\begin{aligned} \vert \nabla u\vert ^{\min \{p(z),q(z)\}+r}\in L^1(Q_T)\quad \text {with any }r\in \left( 0,\frac{4}{N+2}\right) , \end{aligned}$$ | ∇ u | min { p ( z ) , q ( z ) } + r ∈ L 1 ( Q T ) with any r ∈ 0 , 4 N + 2 , which is derived with the help of new interpolation inequalities in the variable Sobolev spaces. The global second-order differentiability of the strong solution is proven: $$\begin{aligned} D_i\left( \sqrt{\mathcal {F}(z,\nabla u)}D_j u\right) \in L^{2}(Q_T),\quad i=1,2,\ldots ,N. \end{aligned}$$ D i F ( z , ∇ u ) D j u ∈ L 2 ( Q T ) , i = 1 , 2 , … , N . The same results are obtained for the equation with the regularized flux $$\mathcal {F}(z,\sqrt{\epsilon ^2+(\xi ,\xi )})\xi $$ F ( z , ϵ 2 + ( ξ , ξ ) ) ξ .
<abstract><p>We consider a parabolic partial differential equation and system derived from a production planning problem dependent on time. Our goal is to find a closed-form solution for the problem considered in our model. Our new theoretical results can be applied in the real world.</p></abstract>
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