# Mathematical Methods in the Applied Sciences

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*L*

^{1}data in anisotropic‐weighted Sobolev spaces of infinite order. Copyright © 2009 John Wiley & Sons, Ltd.

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*W*

^{1, p}(Ω⊂ℝ

^{n}),

*p*>

*n*,

*p*⩾2, was proved by Wrzosek. He also proved that the ω‐limit set consists of regular stationary solutions. In this paper, we prove that...

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*L*

_{1}‐setting with generally boundary conditions. Copyright © 2009 John Wiley & Sons, Ltd.

Mathematical Methods in the Applied Sciences > 33 > 1 > 57 - 70

*a priori*estimates and some density arguments, we prove the well posedness of the associated linear problem. The existence and uniqueness of the weak solution of the nonlinear problem are then established by applying an iterative process based on the obtained results for the linear problem. Copyright...

Mathematical Methods in the Applied Sciences > 33 > 1 > 103 - 116

^{θ}with θ∈(0, γ⊲2], γ>1. The initial data are a perturbation of a corresponding steady solution and continuously contact with vacuum on the free boundary. The obtained results apply for the one‐dimensional Siant–Venant model...

Mathematical Methods in the Applied Sciences > 33 > 1 > 12 - 24

^{n}with

*n*⩾2. It is proved that if ψ(

*u*)/ϕ(

*u*) grows faster than

*u*

^{2/n}as

*u*→∞ and some further technical conditions are fulfilled, then there exist solutions that blow up in either finite or infinite time. Here, the total mass ∫

_{Ω}

*u*(

*x, t*)d

*x*may attain...

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*p*‐Laplacian where

*ϕ*

_{p}(

*s*)=|

*s*|

^{p−2}

*s, p*>1,

*f*is a lower semi‐continuous function. Using the fixed‐point theorem of cone expansion and compression of norm type, the existence of positive solution and infinitely many positive solutions for Sturm–Liouville‐like...

Mathematical Methods in the Applied Sciences > 33 > 2 > 138 - 146

*u*

_{t}=

*f*(

*x, u*

_{x},

*u*

_{xx})

*u*

_{xxx}+

*g*(

*x, u*

_{x},

*u*

_{xx}) via the method of preliminary group classification. This method is a systematic means of analyzing the equation for symmetries. We find explicit forms of

*f*and

*g*, which allow for a larger dimensional Lie algebra of point symmetries. Copyright © 2009 John Wiley & Sons, Ltd.

Mathematical Methods in the Applied Sciences > 33 > 2 > 177 - 187

*H*${\text{\hspace{0.17em}}}_{\text{1}}^{\text{0}}$(Ω) ×

*L*

_{2}(Ω) and

*H*

^{2}(Ω)∩

*H*${\text{\hspace{0.17em}}}_{\text{1}}^{\text{0}}$(Ω) ×

*H*${\text{\hspace{0.17em}}}_{\text{1}}^{\text{0}}$(Ω). Copyright © 2009 John Wiley & Sons, Ltd.

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