Well-posedness problem of an anisotropic parabolic equation

On a weak solution matching up with the double degenerate parabolic equation

“…By the way, since equation (1.1) is isotropic, generalizing the method used in this paper to an anisotropic parabolic equation also seems very interesting. If A(s) = s and b(x, t) = b(x), some progress has been made in [42,43] in recent years.…”

On a Partial Boundary Value Condition of a Porous Medium Equation with Exponent Variable

“…have been brought to the forefront by many more scholars. Since the beginning of this century, there are a great deal of papers devoting to the well-posedness problem, the intrinsic Harnack inequalities, the long-time behavior, and the Hölder regularity of weak solutions, one can refer to the literatures [20][21][22][23][24][25][26][27][28][29][30][31] and the references therein.…”

On the Boundary Value Condition of an Isotropic Parabolic Equation

“…has been studied by the second author in recent years [19][20][21]. Instead of boundary value condition (3), only a partial boundary value condition…”

“…is imposed, where ∑ p ∈ ∂Ω is a relatively open subset which has different expression according to different kinds of f ðx, t, u,∇uÞ and sometimes is just an empty set [19][20][21].…”

“…Compared with [19,20] and [21], since the diffusion coefficient a i ðx, tÞ and the variable exponent p i ðx, tÞ both depend on the time variable t, equation (1) has a wider applications than equation (6), and in mathematical theory, there are some essential difficulties to be overcome. More than that, instead of (5), we only assume that a i x, t ð Þ> 0, x, t ð Þ∈ Ω × 0, T…”