Holder Estimates for Local Solutions of Some Doubly Nonlinear Degenerate Parabolic Equations

Porzio, Maria Michaela; Vespri, Vincenzo

doi:10.1006/jdeq.1993.1045

Cited by 425 publications

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“…Since u , v are Hölder continuous in Q T , bounded in C(Q T ) uniformly in > 0 and the structure conditions of [19] are satisfied for the equations of system (1), whenever ∈ (0, 1/2), Theorem 1.2 of [19] applies to conclude that the following inequality…”

Section: If the Non-trivial Pairmentioning

confidence: 99%

“…Let (u n , v n , a n , b n ) ∈ S be a minimizing sequence for the cost functional J; hence, by the existence results of the previous section, (u n , v n ) ∈ C(Q T ) × C(Q T ), u n , v n ≥ 0 in Q T and u n C(Q T ) , v n C(Q T ) > λ 0 , is a solution of (1) corresponding to (a n , b n ) ∈ U . Now, set u 0 n (·) := u n (·, 0) = u n (·, T ) and v 0 n (·) := v n (·, 0) = v n (·, T ), hence u 0 n , v 0 n are Hölder continuous in Ω by the regularity results of [19], and consider the solution w n ∈ L 2 (0, T ) of the problem ẇ(t) = Ω h n (x, t)dx, a.e. in [0, T ], w(0) = 0,…”

Section: F X T U(x T) V(x T) A(x T) B(x T) Dxdtmentioning

confidence: 99%

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Positive periodic solutions and optimal control for a distributed biological model of two interacting species

Fragnelli

Nistri

Papini

2011

Nonlinear Analysis: Real World Applications

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The paper deals with the existence of positive periodic solutions to a system of degenerate parabolic equations with delayed nonlocal terms and Dirichlet boundary conditions. Taking in each equation a meaningful function as a control parameter, we show that for a suitable choice of a class of such controls we have, for each of them, a time-periodic response of the system under different assumptions on the kernels of the nonlocal terms. Finally, we consider the problem of the minimization of a cost functional on the set of pairs: controlperiodic response. The considered system may be regarded as a possible model for the coexistence problem of two biological populations, which dislike crowding and live in a common territory, under different kind of intra-and inter-specific interferences.

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Section: If the Non-trivial Pairmentioning

confidence: 99%

Section: F X T U(x T) V(x T) A(x T) B(x T) Dxdtmentioning

confidence: 99%

Positive periodic solutions and optimal control for a distributed biological model of two interacting species

Fragnelli

Nistri

Papini

2011

Nonlinear Analysis: Real World Applications

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“…Due to (31) and (32) we can apply the Gagliardo-Nirenberg inequality in the form of Lemma 3.4 i) to obtain c 2 > 0 such that…”

Section: Local Existencementioning

confidence: 99%

“…and hence (31) and (35) enable us to invoke the Gagliardo-Nirenberg inequality and obtain c 4 > 0 such that on (0, T )…”

Section: Local Existencementioning

confidence: 99%

Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion

Lankeit

2017

Journal of Differential Equations

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“…As to particular authors who have contributed to the theory so far, we mention Alt and Luckhaus [1], Atkinson and Bouillet [7,17], Bamberger [8], Bertsch [10][11][12][13], Blanc [14][15][16], Dal Passo [10,11,18], Diaz and de Thelin [19], DiBenedetto [21], van Duijn [12,[22][23][24], Esteban [12,13,25,26], Hilhorst [24], Ivanov [31], Kacur [33,34], Kalashnikov [35][36][37][38][39][40], Lions [45], Nagai [47], Pavlov [48,49], Porzio [51], Rykov [53,54], Tsutsumi [57], Vazquez [25,26], Vespri [51,[58][59][60][61], Yin …”

Section: ) DXmentioning

confidence: 99%

A necessary and sufficient condition for finite speed of propagation in the theory of doubly nonlinear degenerate parabolic equations

Gilding

Kersner

1996

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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A degenerate parabolic partial differential equation with a time derivative and first-and second-order derivatives with respect to one spatial variable is studied. The coefficients in the equation depend nonlinearly on both the unknown and the first spatial derivative of a function of the unknown. The equation is said to display finite speed of propagation if a non-negative weak solution which has bounded support with respect to the spatial variable at some initial time, also possesses this property at later times. A criterion on the coefficients in the equation which is both necessary and sufficient for the occurrence of this phenomenon is established. According to whether or not the criterion holds, weak travelling-wave solutions or weak travelling-wave strict subsolutions of the equation are constructed and used to prove the main theorem via a comparison principle. Applications to special cases are provided.

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Holder Estimates for Local Solutions of Some Doubly Nonlinear Degenerate Parabolic Equations

Cited by 425 publications

References 0 publications

Positive periodic solutions and optimal control for a distributed biological model of two interacting species

Positive periodic solutions and optimal control for a distributed biological model of two interacting species

Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion

A necessary and sufficient condition for finite speed of propagation in the theory of doubly nonlinear degenerate parabolic equations

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