Search

In this paper, we establish the boundary regularity results for viscosity solutions of fully nonlinear degenerate/singular parabolic equations of the form u t - x n γ F (D 2 u , x , t) = f , where γ < 1 . These equations are motivated by the porous media type or fast diffusion type equations. We show the boundary C 1 , α -regularity of functions in their solutions class and the boundary C 2 , α -regularity of viscosity solutions. As an application, we derive the global regularity results and the solvability of the Cauchy–Dirichlet problems. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the following quasilinear chemotaxis–consumption system with degenerate signal-dependent motility u t = ∇ · (v α ∇ u m) + α ∇ · (v α - 1 u ∇ v) + r u - μ u 2 , x ∈ Ω , t > 0 , v t = Δ v - u v , x ∈ Ω , t > 0 , under the homogeneous Neumann boundary condition in Ω ⊂ R n (n ≥ 1) with α , r , μ > 0 and m > 1 . It is shown that above system admits at least one global weak solution fulfilling the following boundedness property ‖ u (· , t) ‖ L p (Ω) + ‖ v (· , t) ‖ W 1 , ∞ (Ω) ≤ C for all t > 0 , p > 2 and m > 1 . This result gives an evident of the regularized effect of porous-medium-type diffusion. [ABSTRACT FROM AUTHOR]

The purpose of the present work is investigating a degenerate parabolic equation with anisotropic p-Laplacian operator and strongly nonlinear source under boundary conditions of Dirichlet type; the existence of a periodic non-negative solution is shown. The proof is based on the Leray–Schauder topological degree, which is tricky to work with in this kind of equations. [ABSTRACT FROM AUTHOR]

In this paper, we show an abstract setting for analyzing the solvability of a class of nonlinear mixed degenerate parabolic equations. Specifically, we combine the well-known theory for stationary mixed problems and the theory for nonlinear degenerate parabolic equations, to obtain sufficient conditions inspired in real-world application problems to guarantee the existence and uniqueness of solution. Furthermore, we illustrate some applications of the abstract framework through particular problems arising from the field of physical electromagnetic sciences. Initially, we introduce two formulations based on a time primitive of the electric field, encompassing both local and nonlocal source conditions. Finally, we explore a nonlinear axisymmetric eddy current model employed in the context of induction furnaces. [ABSTRACT FROM AUTHOR]

We consider the homogeneous Dirichlet problem for the parabolic equation ut−div(∣∇u∣p(x,t)−2∇u)=f(x,t)+F(x,t,u,∇u){u}_{t}-{\rm{div}}({| \nabla u| }^{p\left(x,t)-2}\nabla u)=f\left(x,t)+F\left(x,t,u,\nabla u) in the cylinder QT≔Ω×(0,T){Q}_{T}:= \Omega \times \left(0,T), where Ω⊂RN\Omega \subset {{\mathbb{R}}}^{N}, N≥2N\ge 2, is a C2{C}^{2}-smooth or convex bounded domain. It is assumed that p∈C0,1(Q¯T)p\in {C}^{0,1}\left({\overline{Q}}_{T}) is a given function and that the nonlinear source F(x,t,s,ξ)F\left(x,t,s,\xi ) has a proper power growth with respect to ss and ξ\xi . It is shown that if p(x,t)>2(N+1)N+2p\left(x,t)\gt \frac{2\left(N+1)}{N+2}, f∈L2(QT)f\in {L}^{2}\left({Q}_{T}), ∣∇u0∣p(x,0)∈L1(Ω){| \nabla {u}_{0}| }^{p\left(x,0)}\in {L}^{1}\left(\Omega ), then the problem has a solution u∈C0([0,T];L2(Ω))u\in {C}^{0}\left(\left[0,T];\hspace{0.33em}{L}^{2}\left(\Omega )) with ∣∇u∣p(x,t)∈L∞(0,T;L1(Ω)){| \nabla u| }^{p\left(x,t)}\in {L}^{\infty }\left(0,T;\hspace{0.33em}{L}^{1}\left(\Omega )), ut∈L2(QT){u}_{t}\in {L}^{2}\left({Q}_{T}), obtained as the limit of solutions to the regularized problems in the parabolic Hölder space. The solution possesses the following global regularity properties: ∣∇u∣2(p(x,t)−1)+r∈L1(QT),for any 0

In this paper we prove existence and regularity results for a class of parabolic problems with irregular initial data and lower order terms singular with respect to the solution. We prove that, even if the initial datum is not bounded but only in L 1 (Ω) , there exists a solution that "instantly" becomes bounded. Moreover we study the behavior in time of these solutions showing that this class of problems admits global solutions which all have the same behavior in time independently of the size of the initial data. [ABSTRACT FROM AUTHOR]

We study local boundedness and Hölder continuity of a parabolic equation involving the fractional p-Laplacian of order s, with 0 < s < 1 , 2 ≤ p < ∞ , with a general right-hand side. We focus on obtaining precise Hölder continuity estimates. The proof is based on a perturbative argument using the already known Hölder continuity estimate for solutions to the equation with zero right-hand side. [ABSTRACT FROM AUTHOR]

We prove a local higher integrability result for the gradient of a weak solution to parabolic double-phase systems of p-Laplace type when 2 n n + 2 < p ≤ 2 . The result is based on a reverse Hölder inequality in intrinsic cylinders combining p-intrinsic and (p, q)-intrinsic geometries. A singular scaling deficits affects the range of q. [ABSTRACT FROM AUTHOR]

We study the symmetric stochastic p-Stokes system, p ∈ (1 , ∞) , in a bounded domain. The results are two-fold: First, we show that in the context of analytically weak solutions, the stochastic pressure—related to non-divergence free stochastic forces—enjoys almost - 1 / 2 temporal derivatives on a Besov scale. Second, we verify that the velocity u of strong solutions obeys 1/2 temporal derivatives in an exponential Nikolskii space. Moreover, we prove that the non-linear symmetric gradient V (ϵ u) = (κ + ϵ u) (p - 2) / 2 ϵ u , κ ≥ 0 , which measures the ellipticity of the p-Stokes system, has 1/2 temporal derivatives in a Nikolskii space. [ABSTRACT FROM AUTHOR]

In this paper, we derive a comparison principle for non-negative weak sub- and super-solutions to doubly nonlinear parabolic partial differential equations whose prototype is ∂ t u q - div (| ∇ u | p - 2 ∇ u) = 0 in Ω T , with q > 0 and p > 1 and Ω T : = Ω × (0 , T) ⊂ R n + 1 . Instead of requiring a lower bound for the sub- or super-solutions in the whole domain Ω T , we only assume the lateral boundary data to be strictly positive. The main results yield some applications. Firstly, we obtain uniqueness of non-negative weak solutions to the associated Cauchy–Dirichlet problem. Secondly, we prove that any weak solution is also a viscosity solution. [ABSTRACT FROM AUTHOR]

We consider the initial-boundary value problem for degenerate parabolic and pseudo-parabolic equations associated with Hörmander-type operator. Under the subcritical growth restrictions on the nonlinearity f(u), which are determined by the generalized Métivier index, we establish the global existence of solutions and the corresponding attractors. Finally, we show the upper semicontinuity of the attractors in the topology of H X , 0 1 (Ω) . [ABSTRACT FROM AUTHOR]

We prove the uniform gradient estimates of the nonnegative solution to the Cauchy problem for quasilinear degenerate parabolic equations with nonpower nonlinearity. The large-time estimates in terms of initial data mass and nonlinearity structure were obtained. In the particular case, when the nonlinearity of the equation is power like, then our result coincides with well-known optimal bound. [ABSTRACT FROM AUTHOR]

Given a bounded domain D ⊂ R N and m > 1 , we study the long-time behaviour of solutions to the porous medium equation (PME) posed in a tube ∂ t u = Δ u m in D × R , t > 0 , with homogeneous Dirichlet boundary conditions on the boundary ∂ D × R and suitable initial datum at t = 0 . In two previous works, Vázquez and Gilding & Goncerzewicz proved that a wide class of solutions exhibit a traveling wave behaviour, when computed at a logarithmic time-scale and suitably renormalized. In this paper, we show that, for large times, solutions converge in relative error to the Friendly Giant, i.e., the unique nonnegative solution to the PME posed in the section D of the tube (with homogeneous Dirichlet boundary conditions) having a special self-similar form. In addition, sharp rates of convergence and uniform bounds for the location of the free boundary of solutions are given. [ABSTRACT FROM AUTHOR]

This paper establishes a structure-preserving numerical scheme for the Cahn–Hilliard equation with degenerate mobility. First, by applying a finite volume method with upwind numerical fluxes to the degenerate Cahn–Hilliard equation rewritten by the scalar auxiliary variable (SAV) approach, we creatively obtain an unconditionally bound-preserving, energy-stable and fully-discrete scheme, which, for the first time, addresses the boundedness of the classical SAV approach under H - 1 -gradient flow. Then, a dimensional-splitting technique is introduced in high-dimensional cases, which greatly reduces the computational complexity while preserves original structural properties. Numerical experiments are presented to verify the bound-preserving and energy-stable properties of the proposed scheme. Finally, by applying the proposed structure-preserving scheme, we numerically demonstrate that surface diffusion is approximated by the Cahn–Hilliard equation with degenerate mobility and Flory–Huggins potential, when the absolute temperature is sufficiently low, which agrees well with the theoretical result by using formal asymptotic analysis. [ABSTRACT FROM AUTHOR]

We study a general form of a degenerate or singular parabolic equation u t - | D u | γ (Δ u + (p - 2) Δ ∞ N u) = 0 that generalizes both the standard parabolic p-Laplace equation and the normalized version that arises from stochastic game theory. We develop a systematic approach to study second order Sobolev regularity and show that D 2 u exists as a function and belongs to L loc 2 for a certain range of parameters. In this approach proving the estimate boils down to verifying that a certain coefficient matrix is positive definite. As a corollary we obtain, under suitable assumptions, that a viscosity solution has a Sobolev time derivative belonging to L loc 2 . [ABSTRACT FROM AUTHOR]

This paper deals with the numerical analysis for a family of nonlinear degenerate parabolic problems. The model is spatially discretized using a finite element method; an implicit Euler scheme is employed for time discretization. We deduce sufficient conditions to ensure that the fully discrete problem has a unique solution and to prove quasi-optimal error estimates for the approximation. Finally, we propose a nonlinear degenerate parabolic problem that arises from electromagnetic applications in conductive nonlinear magnetic media and deduce its solubility and convergence by using the developed abstract theory, including some numerical results to confirm the obtained theoretical results. [ABSTRACT FROM AUTHOR]

This paper is devoted to the singular limit of a model for the regulation of growth and patterning in developing tissues by diffusing morphogens. The model is governed by a system of nonlinear PDEs. The arguments are based on energy estimates and a change of variable that reduces the system into a nonlinear PDE with singular diffusion. [ABSTRACT FROM AUTHOR]

This paper is concerned with the migration–consumption taxis system involving signal-dependent motilities u t = Δ (u m ϕ (v)) , v t = Δ v - u v , (∗) in smoothly bounded domains Ω ⊂ R n , where m > 1 and n ≥ 2 . It is shown that if ϕ ∈ C 3 ([ 0 , ∞)) is strictly positive on [ 0 , ∞) , for all suitably regular initial data an associated no-flux type initial-boundary value problem possesses a globally defined bounded weak solution, provided m > n 2 , which is consistent with the restriction imposed on m in corresponding signal production counterparts of (⋆) so as to establish the similar result. [ABSTRACT FROM AUTHOR]

We prove the continuous dependence, with respect to the initial datum of solutions of the "quenching parabolic problem" ∂ t u - Δ u + χ { u > 0 } u - β = λ u p , with zero Dirichlet boundary conditions, when β ∈ (0 , 1) , p ∈ (0 , 1 ] , λ ≥ 0 and χ { u > 0 } denotes the characteristic function of the set of points (x, t) where u (x , t) > 0 . Notice that the absorption term χ { u > 0 } u - β is singular and monotone decreasing which does not allow the application of standard monotonicity arguments. [ABSTRACT FROM AUTHOR]

We study the homogeneous Dirichlet problem for a class of nonlinear anisotropic parabolic double phase equations with nonstandard growth conditions. We prove the existence of a unique nonnegative weak solution in suitable Orlicz-Sobolev spaces, and derive the global boundedness of the weak solution. [ABSTRACT FROM AUTHOR]

We study the homogeneous Dirichlet problem for the parabolic equations ut−div(A(z,∣∇u∣)∇u)=F(z,u,∇u),z=(x,t)∈Ω×(0,T),{u}_{t}-{\rm{div}}\left({\mathcal{A}}\left(z,| \nabla u| )\nabla u)=F\left(z,u,\nabla u),\hspace{1.0em}z=\left(x,t)\in \Omega \times \left(0,T), with the double phase flux A(z,∣∇u∣)∇u=(∣∇u∣p(z)−2+a(z)∣∇u∣q(z)−2)∇u{\mathcal{A}}\left(z,| \nabla u| )\nabla u=(| \nabla u{| }^{p\left(z)-2}+a\left(z)| \nabla u{| }^{q\left(z)-2})\nabla u and the nonlinear source FF. The initial function belongs to a Musielak-Orlicz space defined by the flux. The functions aa, pp, and qq are Lipschitz-continuous, a(z)a\left(z) is nonnegative, and may vanish on a set of nonzero measure. The exponents pp, and qq satisfy the balance conditions 2NN+20\varepsilon \gt 0.

We study the stochastic p-Laplace system in a bounded domain. We propose two new space–time discretizations based on the approximation of time-averaged values. We establish linear convergence in space and 1/2 convergence in time. Additionally, we provide a sampling algorithm to construct the necessary random input in an efficient way. The theoretical error analysis is complemented by numerical experiments. [ABSTRACT FROM AUTHOR]

We prove the qualitative decay in time estimate of supremum norm of the nonnegative solution to the Cauchy problem for quasilinear degenerate parabolic equations with nonpower nonlinearity. The result obtained is given in terms of the initial data mass and nonlinearity structure. In particular, if the nonlinearity of the equation is power like, then our result coincides with well-known optimal bound. [ABSTRACT FROM AUTHOR]

In this paper, we study the sharp interface limit for solutions of the Cahn–Hilliard equation with disparate mobilities. This means that the mobility function degenerates in one of the two energetically favorable configurations, suppressing the diffusion in that phase. First, we construct suitable weak solutions to this Cahn–Hilliard equation. Second, we prove precompactness of these solutions under natural assumptions on the initial data. Third, under an additional energy convergence assumption, we show that the sharp interface limit is a distributional solution to the Hele–Shaw flow with optimal energy-dissipation rate. [ABSTRACT FROM AUTHOR]

Existence of non-negative weak solutions is shown for a full curvature thin-film model of a liquid thin film flowing down a vertical fibre. The proof is based on the application of a priori estimates derived for energy-entropy functionals. Long-time behaviour of these weak solutions is analysed and, under some additional constraints for the model parameters and initial values, convergence towards a travelling wave solution is obtained. Numerical studies of energy minimisers and travelling waves are presented to illustrate analytical results. [ABSTRACT FROM AUTHOR]

The present analysis provides insights and analytical findings for a cooperative system formulated with non-linear diffusion. This work discusses, from an analytical perspective, the interaction of two species when both cooperate to explore new territories. Firstly, the analysis focuses on regularity, existence and uniqueness of weak solutions considering slightly positivity in the non-linear operator together with a monotone argument in the independent terms. Afterwards, analytical solution profiles are provided making use of selfsimilar solutions. Non-linear diffusion induces the existence of a propagating front and determines the property of finite speed of propagation in both species when they cooperate to explore new territories. [ABSTRACT FROM AUTHOR]

We study an anisotropic, possibly non-homogeneous version of the evolution 𝑝-Laplacian equation when fast diffusion holds in all directions. We develop the basic theory and prove symmetrization results from which we derive sharp L1L^{1}-L∞L^{\infty} estimates. We prove the existence of a self-similar fundamental solution of this equation in the appropriate exponent range, and uniqueness in a smaller range. We also obtain the asymptotic behaviour of finite mass solutions in terms of the self-similar solution. Positivity, decay rates as well as other properties of the solutions are derived. The combination of self-similarity and anisotropy is not common in the related literature. It is however essential in our analysis and creates mathematical difficulties that are solved for fast diffusions.

We study the comparison principle for non-negative solutions of the equation ∂ ⁡ ( | v | p - 2 ⁢ v) ∂ ⁡ t = div ⁡ ( | ∇ ⁡ v | p - 2 ⁢ ∇ ⁡ v) , 1 < p < ∞. This equation is related to extremals of Poincaré inequalities in Sobolev spaces. We apply our result to obtain pointwise control of the large time behavior of solutions. [ABSTRACT FROM AUTHOR]

We establish sharp criteria for the instantaneous propagation of free boundaries in solutions to the thin-film equation. The criteria are formulated in terms of the initial distribution of mass (as opposed to previous almost-optimal results), reflecting the fact that mass is a locally conserved quantity for the thin-film equation. In the regime of weak slippage, our criteria are at the same time necessary and sufficient. The proof of our upper bounds on free boundary propagation is based on a strategy of "propagation of degeneracy" down to arbitrarily small spatial scales: We combine estimates on the local mass and estimates on energies to show that "degeneracy" on a certain space-time cylinder entails "degeneracy" on a spatially smaller space-time cylinder with the same time horizon. The derivation of our lower bounds on free boundary propagation is based on a combination of a monotone quantity and almost optimal estimates established previously by the second author with a new estimate connecting motion of mass to entropy production. [ABSTRACT FROM AUTHOR]

The intention along this study is to provide an analytical approach to model a flame propagation formulated with a non-linear diffusion. The proposed model can be used to characterize the flame behaviour, and has been previously discussed with a linear diffusion with accurate results. Nonetheless, along this analysis, the non-linear diffusion is introduced supported by the mathematical properties induced by the porous medium equation theory formulated within the partial differential equations theory. First of all, regularity, existence and uniqueness of solutions are shown to hold. Afterwards, asymptotic solutions are obtained with purely analytical techniques. Particular solutions for the flame pressure and temperature variables are provided. Aiming the search of travelling waves (TW) profiles, a minimal travelling wave speed is provided to ensure a purely monotone travelling wave behaviour free of oscillation. In addition, the geometric perturbation theory is explored to show the normal hyperbolic condition of the manifolds used to obtain the travelling waves profiles. The existence of such profiles are based on a topological argument in the phase plane given by the involved manifolds close the critical point. Afterwards, dedicated travelling wave profiles are obtained showing an exponential behaviour and the existence of a propagating support. Note that the existence of an exponential solution is not trivial in the non-linear diffusion case. Finally, the property known as finite speed of propagation is shown to exist and a precise supersolution has been obtained to model the flame propagation at the beginning of the process. The finite speed precludes the existence of an evolving flame front that has been precisely determined. [ABSTRACT FROM AUTHOR]

We study monotone P1 finite element methods on unstructured meshes for fully non-linear, degenerately parabolic Isaacs equations with isotropic diffusions arising from stochastic game theory and optimal control and show uniform convergence to the viscosity solution. Elliptic projections are used to manage singular behaviour at the boundary and to treat a violation of the consistency conditions from the framework by Barles and Souganidis by the numerical operators. Boundary conditions may be imposed in the viscosity or in the strong sense, or in a combination thereof. The presented monotone numerical method has well-posed finite dimensional systems, which can be solved efficiently with Howard’s method. [ABSTRACT FROM AUTHOR]

Under a sharp asymptotic growth condition at infinity, we prove a Liouville type theorem for the inhomogeneous porous medium equation, provided it stays universally close to the heat equation. Additionally, for the homogeneous equation, we show that for the conclusion to hold, it is enough to assume the sharp asymptotic growth at infinity only in the space variable. The results are optimal, meaning that the growth condition at infinity cannot be weakened. [ABSTRACT FROM AUTHOR]

We consider nonnegative solutions to the Cauchy problem for a degenerate parabolic system of the form u t = div ⁡ (v α 1 ∇ u m 1 ) , (x , t) ∈ S T = R N × (0 , T) v t = div ⁡ (u α 2 ∇ v m 2 ) , (x , t) ∈ S T u (x , 0) = u 0 (x) ≥ 0 , v (x , 0) = v 0 (x) ≥ 0 , x ∈ R N , N ≥ 1. Under the suitable assumptions on the parameters of nonlinearities and initial data, we obtained optimal decay estimates of a solution for a large time. Moreover, the phenomena of extinction in finite time was established. Provided that initial data are compactly supported, we proved the property of finite speed of propagation. [ABSTRACT FROM AUTHOR]

We consider the high-dimensional equation ∂t⁡u-Δ⁢um+u-β⁢χ{u>0}=0{\partial_{t}u-\Delta u^{m}+u^{-\beta}{\chi_{\{u>0\}}}=0}, extending the mathematical treatment made in 1992 by B. Kawohl and R. Kersner for the one-dimensional case. Besides the existence of a very weak solution u∈𝒞⁢([0,T];Lδ1⁢(Ω)){u\in\mathcal{C}([0,T];L_{\delta}^{1}(\Omega))}, with u-β⁢χ{u>0}∈L1⁢((0,T)×Ω){u^{-\beta}\chi_{\{u>0\}}\in L^{1}((0,T)\times\Omega)}, δ⁢(x)=d⁢(x,∂⁡Ω){\delta(x)=d(x,\partial\Omega)}, we prove some pointwise gradient estimates for a certain range of the dimension N, m≥1{m\geq 1} and β∈(0,m){\beta\in(0,m)}, mainly when the absorption dominates over the diffusion (1≤m

In this paper, we consider a liquid–gas system with two components: water and hydrogen flow model in heterogeneous porous media with periodic microstructure taking into account kinetics in the mass transfer between the two phases. The particular feature in this model is that chemistry effects are taken into account. The microscopic model consists of the usual equations derived from the mass conservation laws of both fluids, along with the Darcy–Muskat and capillary pressure laws and the mass exchange is modeled as a source term in the equations. The problem is written in the terms of the phase formulation; i.e. the saturation of one phase, the pressure of the second phase, and the concentration of dissolved hydrogen in the liquid phase. The mathematical model consists in a system of partial differential equations: two degenerate nonlinear parabolic equations and one diffusion–convection equation. The major difficulties related to this new model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we obtain a nonlinear homogenized coupled system of three coupled partial differential equations with effective coefficients which are computed via solving cell problems. We give a rigorous mathematical derivation of the effective model by means of the two-scale convergence. [ABSTRACT FROM AUTHOR]

In this article we prove a Harnack inequality for non-negative weak solutions to doubly nonlinear parabolic equations of the form ∂ t u - div A (x , t , u , D u m) = div F , where the vector field A fulfills p-ellipticity and growth conditions. We treat the slow diffusion case in its full range, i.e. all exponents m > 0 and p > 1 with m (p - 1) > 1 are included in our considerations. [ABSTRACT FROM AUTHOR]

We study a parabolic equation for the fractional p-Laplacian of order s, for p ≥ 2 and 0 < s < 1 . We provide space-time Hölder estimates for weak solutions, with explicit exponents. The proofs are based on iterated discrete differentiation of the equation in the spirit of Moser's technique. [ABSTRACT FROM AUTHOR]

In this paper we address the weak Radon measure-valued solutions associated with the Young measure for a class of nonlinear parabolic equations with initial data as a bounded Radon measure. This problem is described as follows: { u t = α u x x + β [ φ (u) ] x x + f (u) in Q : = Ω × (0 , T) , u = 0 on ∂ Ω × (0 , T) , u (x , 0) = u 0 (x) in Ω , where T > 0 , Ω ⊂ R is a bounded interval, u 0 is nonnegative bounded Radon measure on Ω, and α , β ≥ 0 , under suitable assumptions on φ and f. In this work we prove the existence and the decay estimate of suitably defined Radon measure-valued solutions for the problem mentioned above. In particular, we study the asymptotic behavior of these Radon measure-valued solutions. [ABSTRACT FROM AUTHOR]

Bacterial biofilms can lead to chronic infections, increase tolerance to antibiotics and disinfectants, resistance to phagocytosis, and other components of the body's immune system. Biofilm formation is implicated in the persistence of staphylococcal infections and chronic Pseudomonas aeruginosa lung infections in cystic fibrosis (CF) patients (which can result from biofilm-growing mucoid strains). Conventional treatments utilize aggressive antibiotic prophylaxis/therapy to prevent/eliminate biofilms, followed by chronic suppressive therapy. Recently, the use of enzymes to dissolve the biofilm matrix was investigated, in addition to quorum sensing inhibitors to increase biofilm susceptibility to antibiotics. Here, we propose a novel strategy, utilizing ultrasound-induced inertial cavitation, to increase antibiotic efficacy. The wall shear stress at the biofilm interface is calculated, and viscoplastic constitutive equations are used to examine the biofilm response to the mechanical stress. Our simulations suggest that the maximum biofilm detachment occurs at high pressure/low frequency, and the mechanical disruption can affect the biochemical processes inside the biofilm resulting in vulnerability to antibiotics. [ABSTRACT FROM AUTHOR]

We consider the first mixed problem for a certain class of anisotropic parabolic equations of the form β x u t ′ − div a t x u ∇ u − b t x u ∇ u = u where μ is a measure and the coefficients contain nonpower nonlinearities in the cylindrical domain DT = (0, T)×Ω, where Ω ⊂ ℝn is a bounded domain. We prove the existence of a renormalized solution of the problem for gt = 0 and a function β(x, r), which increases with respect to r and satisfies the Carath´eodory condition. [ABSTRACT FROM AUTHOR]