A nonlinear diffusion problem with convection and anisotropic nonstandard growth conditions

The nonnegative weak solution of a degenerate parabolic equation with variable exponent growth order

“…is dispensable. If g = 0, equation (1.1) arises from the branches of flows of electro-rheological or thermo-rheological fluids (see [1][2][3]), and the processing of digital images [4][5][6][7][8][9][10][11][12][13][14][15]. If the variable exponent p(x, t) is replaced by a constant p, equation (1.1) becomes the well-known non-Newtonian polytropic filtration equation with orientated convection [16], as well as the convection-diffusion-reaction equation in which the variable can be interpreted as temperature for heat transfer problems, concentration for dispersion problems, etc.…”

“…It is worth pointing out that the requirement on p t (x, t) is only negative and integrable, which is a weaker condition than the corresponding conditions appearing in other papers. Recently, Liu and Dong [15] generalized [14]'s result to a more general equation…”

“…In recent years, we have been interested in the well-posedness of weak solutions to the nonlinear equation 4) with some restrictions in f (x, t, v, ∇v). Different from other researchers' works [7][8][9][10][11][12][13][14][15], in which b(x) = 1 or b(x) > b -> 0, where b -= min x∈Ω b(x), we only assumed that…”

The weak solutions of a doubly nonlinear parabolic equation related to the $p(x)$-Laplacian

“…The evolutionary p(x)-Laplacian equation u t = div |∇u| p(x)-2 ∇u , (x, t) ∈ Q T = Ω × (0, T), (1.1) with the initial value u| t=0 = u 0 (x), x ∈ Ω, (1.2) and the homogeneous boundary value u| Γ T = 0, (x, t) ∈ Γ T = ∂Ω × (0, T), (1.3) has been subject of a profound study from the beginning of this century [1][2][3][4][5][6][7][8][9], where Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω, p(x) is a measurable function.…”

Existence of weak solutions to a certain homogeneous parabolic Neumann problem involving variable exponents and cross-diffusion

“…Additionally, in [8] it is shown that the solutions of a similar problem may vanish in finite time even if the equation combines the directions of slow and fast diffusion, and the extinction moment is estimated in terms of the data. Further, very recently the existence of weak solutions to a homogeneous Dirichlet problem of a nonlinear diffusion equation involving anisotropic variable exponents and convection was studied in [39].…”