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Intrinsic Harnack’s Inequality for a General Nonlinear Parabolic Equation in Non-divergence Form
We prove the intrinsic Harnack’s inequality for a general form of a parabolic equation that generalizes both the standard parabolic p -Laplace...
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A boundary estimate for singular parabolic diffusion equations
We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to singular parabolic...
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Dirichlet Boundary Conditions for Degenerate and Singular Nonlinear Parabolic Equations
We study existence and uniqueness of solutions to a class of quasilinear degenerate parabolic equations, in bounded domains. We show that there...
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Null controllability in large time of a parabolic equation involving the Grushin operator with an inverse-square potential
We prove the null controllability in large time of the following linear parabolic equation involving the Grushin operator with an inverse-square...
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Interior Harnack Estimates: The State-of-the-Art for Quasilinear Singular Parabolic Equations
In this paper we give some historical information about elliptic and parabolic Harnack inequalities. Then we state the main results known for Harnack...
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Viscosity solutions to a parabolic inhomogeneous equation associated with infinity Laplacian
We obtain the existence and uniqueness results of viscosity solutions to the initial and boundary value problem for a nonlinear degenerate and...
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Global existence and uniqueness for the inhomogeneous 1-Laplace evolution equation
In this paper we introduce a new approach to the Dirichlet problem for the total variation flow in a bounded domain and analyze the associated...
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On uniqueness of solutions to the Cauchy problem for degenerate Fokker–Planck–Kolmogorov equations
We prove a new uniqueness result for highly degenerate second-order parabolic equations on the whole space. A novelty is also our class of solutions...
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Almost classical solutions to the total variation flow
The paper examines the one-dimensional total variation flow equation with Dirichlet boundary conditions. Thanks to a new concept of “almost...
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Stability analysis of asymptotic profiles for sign-changing solutions to fast diffusion equations
Every solution u = u ( x , t ) of the Cauchy–Dirichlet problem for the fast diffusion equation, ∂ t (| u | m -2 u ) = Δu in Ω × (0, ∞) with a smooth bounded...
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On well-posedness of the semilinear heat equation on the sphere
We are concerned with existence, uniqueness and nonuniqueness of nonnegative solutions to the semilinear heat equation in open subsets of the n -dimens...
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The fast diffusion equation with measure data
We show the existence of solutions to the fast diffusion equation with a general finite and positive Borel measure as the right hand side source term.
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Well-posedness of the Cauchy problem for nonlinear parabolic equations with variable density in the hyperbolic space
We investigate the well-posedness of the Cauchy problem for a class of nonlinear parabolic equations with variable density in the hyperbolic space....
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Hölder continuity for Trudinger’s equation in measure spaces
We complete the study of the regularity for Trudinger’s equation by proving that weak solutions are Hölder continuous also in the singular case. The...
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Uniqueness and support properties of solutions to singular quasilinear parabolic equations on surfaces of revolution
We study uniqueness, nonuniqueness and support properties of nonnegative bounded solutions of initial value problems on surfaces of revolution with...