Strong solutions of evolution equations with p(x,t)-Laplacian: Existence, global higher integrability of the gradients and second-order regularity
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Cited by 10 publications
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We study the homogeneous Dirichlet problem for the equationwhere Ω ⊂ R N , N ≥ 2, is a bounded domain with ∂Ω ∈ C 2 . The variable exponents p, q and the nonnegative modulating coefficients a, b are given Lipschitz-continuous functions of the argument z = (x, t) ∈ Q T . It is assumed that 2N N+2 < p(z), q(z) and that the modulating coefficients and growth exponents satisfy the balance conditionsWe find conditions on the source f and the initial data u(•, 0) that guarantee the existence of a unique strong solution u with ut ∈ L 2 (Q T ) and a|∇u| p + b|∇u| q ∈ L ∞ (0, T ; L 1 (Ω)). The solution possesses the property of global higher integrability of the gradient, |∇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 secondorder differentiability of the strong solution is proven:Dx i a|∇u| p−2 + b|∇u| q−2 1 2 Dx j u ∈ L 2 (Q T ), i, j = 1, 2, . . . , N.1,p(•) 0
We study the homogeneous Dirichlet problem for the equationwhere Ω ⊂ R N , N ≥ 2, is a bounded domain with ∂Ω ∈ C 2 . The variable exponents p, q and the nonnegative modulating coefficients a, b are given Lipschitz-continuous functions of the argument z = (x, t) ∈ Q T . It is assumed that 2N N+2 < p(z), q(z) and that the modulating coefficients and growth exponents satisfy the balance conditionsWe find conditions on the source f and the initial data u(•, 0) that guarantee the existence of a unique strong solution u with ut ∈ L 2 (Q T ) and a|∇u| p + b|∇u| q ∈ L ∞ (0, T ; L 1 (Ω)). The solution possesses the property of global higher integrability of the gradient, |∇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 secondorder differentiability of the strong solution is proven:Dx i a|∇u| p−2 + b|∇u| q−2 1 2 Dx j u ∈ L 2 (Q T ), i, j = 1, 2, . . . , N.1,p(•) 0
We study the homogeneous Dirichlet problem for the evolution p(x, t)-Laplacian with the nonlinear source $$\begin{aligned} u_t-{\text {div}}\left( |\nabla u|^{p(x,t)-2}\nabla u\right) =f(x,t,u),\quad (x,t)\in Q=\Omega \times (0,T). \end{aligned}$$ u t - div | ∇ u | p ( x , t ) - 2 ∇ u = f ( x , t , u ) , ( x , t ) ∈ Q = Ω × ( 0 , T ) . Here, $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ R n is a bounded domain, $$n\ge 2$$ n ≥ 2 , and $$p(x,\!t)$$ p ( x , t ) is a given function $$p(\cdot ):Q\mapsto (\frac{2n}{n+2},p^+]$$ p ( · ) : Q ↦ ( 2 n n + 2 , p + ] , $$p^+<\infty $$ p + < ∞ . It is shown that the solution is stable with respect to perturbations of the exponent p(x, t), the nonlinear source f(x, t, u), and the initial datum. We obtain quantitative estimates on the norm of the difference between two solutions in a variable Sobolev space through the norms of perturbations of the exponent p(x, t) and the data u(x, 0), f. Estimates on the rate of convergence of solutions of perturbed problems to the solution of the limit problem are derived.
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 .
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