$L^\infty $ estimates of solution for $m$-Laplacian parabolic equation with a nonlocal term

Pulun Hou; Caisheng Chen

Displaying similar documents to “ $L^{\infty}$ estimates of solution for $m$ -Laplacian parabolic equation with a nonlocal term”

Asymptotic behavior of positive solutions of a Dirichlet problem involving supercritical nonlinearities

Giovanni Anello, Giuseppe Rao (2013)

Commentationes Mathematicae Universitatis Carolinae

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Let $p > 1$ , $q > p$ , $λ > 0$ and $s \in] 1, p [$ . We study, for $s \to p^{-}$ , the behavior of positive solutions of the problem $- Δ_{p} u = λ u^{s - 1} + u^{q - 1}$ in $Ω$ , $u_{∣ \partial Ω} = 0$ . In particular, we give a positive answer to an open question formulated in a recent paper of the first author.

Global Attractors for a Class of Semilinear Degenerate Parabolic Equations on $ℝ^{N}$

Cung The Anh, Le Thi Thuy (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove the existence of global attractors for the following semilinear degenerate parabolic equation on $ℝ^{N}$ : ∂u/∂t - div(σ(x)∇ u) + λu + f(x,u) = g(x), under a new condition concerning the variable nonnegative diffusivity σ(·) and for an arbitrary polynomial growth order of the nonlinearity f. To overcome some difficulties caused by the lack of compactness of the embeddings, these results are proved by combining the tail estimates method and the asymptotic a priori estimate method. ...

A population biological model with a singular nonlinearity

Sayyed Hashem Rasouli (2014)

Applications of Mathematics

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We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form $\{\begin{matrix} - {div (| x |}^{- α p} {| \nabla u |}^{p - 2} {\nabla u) = | x |}^{- (α + 1) p + β} (a u^{p - 1} - f (u) - \frac{c}{u^{γ}}), x \in Ω, \\ u = 0, x \in \partial Ω, \end{matrix}$ where $Ω$ is a bounded smooth domain of $ℝ^{N}$ with $0 \in Ω$ , $1 < p < N$ , $0 \leq α < (N - p) / p$ , $γ \in (0, 1)$ , and $a$ , $β$ , $c$ and $λ$ are positive parameters. Here $f : [0, \infty) \to ℝ$ is a continuous function. This model arises in the studies of population biology of one species with $u$ representing the concentration of the species. We discuss the existence of a positive solution when $f$ satisfies certain additional conditions. We use the method of sub-supersolutions...

Absence of global solutions to a class of nonlinear parabolic inequalities

M. Guedda (2002)

Colloquium Mathematicae

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We study the absence of nonnegative global solutions to parabolic inequalities of the type $u_{t} \geq - {(- Δ)}^{β / 2} u - V (x) u + h (x, t) u^{p}$ , where ${(- Δ)}^{β / 2}$ , 0 < β ≤ 2, is the β/2 fractional power of the Laplacian. We give a sufficient condition which implies that the only global solution is trivial if p > 1 is small. Among other properties, we derive a necessary condition for the existence of local and global nonnegative solutions to the above problem for the function V satisfying ${V ₊ (x) \sim a | x |}^{- b}$ , where a ≥ 0, b > 0, p > 1 and V₊(x): = maxV(x),0....

Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities

Philippe Souplet, Slim Tayachi (2001)

Colloquium Mathematicae

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Consider the nonlinear heat equation (E): $u_{t} - Δ u = {| u |}^{p - 1} u + b {| \nabla u |}^{q}$ . We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates $C ₁ {(T - t)}^{- 1 / (p - 1)} {\leq | | u (t) | |}_{\infty} \leq C ₂ {(T - t)}^{- 1 / (p - 1)}$ . Also, as an application of our method, we obtain the same upper estimate if u only satisfies the nonlinear parabolic inequality $u_{t} - u_{x x} \geq u^{p}$ . More general inequalities of the form $u_{t} - u_{x x} \geq f (u)$ with, for instance, $f (u) = (1 + u) l o g^{p} (1 + u)$ are also treated. Our results show that for solutions of the parabolic inequality, one has essentially the same estimates as for solutions...

On the long-time behaviour of solutions of the p-Laplacian parabolic system

Paweł Goldstein (2008)

Colloquium Mathematicae

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Convergence of global solutions to stationary solutions for a class of degenerate parabolic systems related to the p-Laplacian operator is proved. A similar result is obtained for a variable exponent p. In the case of p constant, the convergence is proved to be $¹_{l o c}$ , and in the variable exponent case, L² and $W^{1, p (x)}$ -weak.

Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $ℝ^{n}$

Alessandra Lunardi (1998)

Studia Mathematica

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We study existence, uniqueness, and smoothing properties of the solutions to a class of linear second order elliptic and parabolic differential equations with unbounded coefficients in $ℝ^{n}$ . The main results are global Schauder estimates, which hold in spite of the unboundedness of the coefficients.

Existence results for a class of nonlinear parabolic equations with two lower order terms

Ahmed Aberqi, Jaouad Bennouna, M. Hammoumi, Mounir Mekkour, Ahmed Youssfi (2014)

Applicationes Mathematicae

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We investigate the existence of renormalized solutions for some nonlinear parabolic problems associated to equations of the form ⎧ $\partial (e^{β u} - 1) / \partial t - {d i v (| \nabla u |}^{p - 2} \nabla u) + d i v (c (x, t) {| u |}^{s - 1} u) + b (x, t) {| \nabla u |}^{r} = f$ in Q = Ω×(0,T), ⎨ u(x,t) = 0 on ∂Ω ×(0,T), ⎩ $(e^{β u} - 1) (x, 0) = (e^{β u ₀} - 1) (x)$ in Ω. with s = (N+2)/(N+p) (p-1), $c (x, t) \in {(L^{τ} (Q T))}^{N}$ , τ = (N+p)/(p-1), r = (N(p-1) + p)/(N+2), $b (x, t) \in L^{N + 2, 1} (Q T)$ and f ∈ L¹(Q).

The initial value problem for parabolic equations with data in $L^{p} (R^{n})$

Eugene Fabes (1972)

Studia Mathematica

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On $L^{p}$ -estimates for solutions of the Cauchy problem for parabolic differential equations

P. Besala (1971)

Annales Polonici Mathematici

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Displaying similar documents to “ L ∞ estimates of solution for m -Laplacian parabolic equation with a nonlocal term”

Displaying similar documents to “ $L^{\infty}$ estimates of solution for $m$ -Laplacian parabolic equation with a nonlocal term”