Existence of very weak solutions of doubly nonlinear parabolic equations with measure data

Abstract

Abstract. We deal with a Cauchy-Dirichlet problem with homogeneous boundary conditions on the parabolic boundary of a space-time cylinder for doubly nonlinear parabolic equations, whose prototype iswith a non-negative Radon measure µ on the right-hand side. Here, the doubly degenerate (p ≥ 2, m ≥ 1) and singular-degenerate (p ∈ ( 2n n+2 , 2), m ≥ 1) cases are considered. The central objective is to establish the existence of a solution in the sense of distributions (see Theorem 1.4). The constructed solution is obtained by a limit of approximations, i.e. we use solutions of regularized Cauchy-Dirichlet problems and pass to the limit to receive a solution for the original CauchyDirichlet problem.

Pointwise estimates via parabolic potentials for a class of doubly nonlinear parabolic equations with measure data

“…Using u α+1 in the regularity assumptions (see Remark 1.2), the existence of less regular very weak solutions to the CauchyDirichlet problem (1.6) was established in [39] under the additional monotonicity condition…”

“…one can prove the existence of weak solutions in the sense of (1.9) (see [39,Rem. 4.3]), whereas [38] supplies the existence of weak solutions in the sense of Definition 1.1, provided that μ ∈ Ls(E T , R ≥0 ) for…”

“…What is more, [33,35,40] are concerned with the asymptotic behavior of solutions to doubly nonlinear parabolic equations for certain values of the quantity p + m, and the local boundedness of the gradient of a solution to the homogeneous equation was shown in [36] under the additional assumption that u is strictly positive. Existence and uniqueness results for the Cauchy-Dirichlet problem with an inhomogeneity μ ∈ L ∞ (E T , R ≥0 ) were developed in [18][19][20] and generalized in [38,39] to Lebesgue integrable functions and Radon measures as right-hand sides.…”

“…For the interested reader, we remark that, apart from our definition of weak solutions, which is also employed in [15,19,33,35,39,42,44], for instance, there is another concept of weak solutions in the context of doubly nonlinear parabolic equations (see [4,10,18,38]), where our regularity assumptions (1.7) are replaced by…”

Geometric estimates for doubly nonlinear parabolic PDEs

“…[15, chapter 2, section 1] and [20, proposition 3.1]), as well as it defines the fashion in which weak solutions enjoy a genuine Hölder modulus of continuity. Now, we comment on literature related to doubly degenerate PDEs: it is worth highlighting that, under structural assumptions (P1)-(P3) and in view of the compatibility conditions (W-CC), existence and uniqueness of solutions in appropriate Sobolev spaces were established in [30,36,37]. Regularity and local behaviour of weak solutions of doubly degenerate evolution models (1.1) received an increasing focus of attention in the last decades (cf [8, 21, 24-29, 35, 39]) due to their intrinsic connection to several problems arising nonlinear potential theory, non-Newtonian fluids, mathematical physics, etc (cf [1,7,16,45] for complete essays on regularity of evolution equations with degenerate diffusion, and [33,38] for pioneering works concerning parabolic potential estimates).…”