Weak solutions to a class of signal-dependent motility Keller-Segel systems with superlinear damping

Abstract In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter vanishes (which turns the problem into a purely nonlocal one) or goes to infinity (reducing the problem to the classical semilinear Laplace equation) is also investigated.

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Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

Stochastic Partial Differential Equations Analysis and Computations ◽

10.1007/s40072-021-00199-6 ◽

2021 ◽

Author(s):

Shohei Nakajima

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Weak Solutions ◽

Stochastic Partial Differential Equations ◽

Fractional Laplacian ◽

Existence Of Solutions ◽

Partial Differential ◽

Continuity Theorem ◽

Existence Of Weak Solutions ◽

One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

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Equivalence of viscosity and weak solutions for a p-parabolic equation

Journal of Evolution Equations ◽

10.1007/s00028-020-00666-y ◽

2021 ◽

Cited By ~ 1

Author(s):

Jarkko Siltakoski

Keyword(s):

Parabolic Equation ◽

Weak Solutions ◽

Lower Semicontinuity ◽

Lower Semicontinuous ◽

Relationship Of ◽

The Relationship

AbstractWe study the relationship of viscosity and weak solutions to the equation $$\begin{aligned} \smash {\partial _{t}u-\varDelta _{p}u=f(Du)}, \end{aligned}$$ ∂ t u - Δ p u = f ( D u ) , where $$p>1$$ p > 1 and $$f\in C({\mathbb {R}}^{N})$$ f ∈ C ( R N ) satisfies suitable assumptions. Our main result is that bounded viscosity supersolutions coincide with bounded lower semicontinuous weak supersolutions. Moreover, we prove the lower semicontinuity of weak supersolutions when $$p\ge 2$$ p ≥ 2 .

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$L^{\infty }$ decay estimates of solutions of nonlinear parabolic equation

Boundary Value Problems ◽

10.1186/s13661-020-01480-8 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Hui Wang ◽

Caisheng Chen

Keyword(s):

Parabolic Equation ◽

Weak Solutions ◽

Nonlinear Parabolic Equation ◽

Divergence Form ◽

Decay Estimates ◽

Laplacian Equation ◽

Estimates Of Solutions ◽

Nonlinear Parabolic ◽

Doubly Nonlinear ◽

Doubly Nonlinear Parabolic Equation

AbstractIn this paper, we are interested in $L^{\infty }$ L ∞ decay estimates of weak solutions for the doubly nonlinear parabolic equation and the degenerate evolution m-Laplacian equation not in the divergence form. By a modified Moser’s technique we obtain $L^{\infty }$ L ∞ decay estimates of weak solutiona.

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A Boundary Estimate for Singular Sub-Critical Parabolic Equations

International Mathematics Research Notices ◽

10.1093/imrn/rnaa351 ◽

2020 ◽

Author(s):

Ugo Gianazza ◽

Naian Liao

Keyword(s):

Modulus Of Continuity ◽

Weak Solutions ◽

Parabolic Equations ◽

Boundary Point ◽

Cylindrical Domain ◽

Boundary Estimate ◽

Critical Range ◽

Type Integral ◽

Wiener Type ◽

Singular Parabolic Equations

Abstract We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to singular parabolic equations of $p$-Laplacian type, with $p$ in the sub-critical range $\big(1,\frac{2N}{N+1}\big]$. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic $p$-capacity.

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