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Results tagged with ap.analysis-of-pdes
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user 33849
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
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maximum principle for a non-uniformly parabolic operator
Smoothness on $G$ may not be enough to ensure that a maximum principle can be obtained. If $G$ or $G_x$ blows up as $|x|\to\infty$ then I anticipate a problem. I advise looking at Protter and Weinberg …
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282
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Examples of non-uniqueness in reaction-diffusion equations
Consider the problem of finding a bounded classical solution $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ (such that $u$ is continuous and $u_t$, $u_x$ and $u_{xx}$ exist and are continuous on $\mathbb{R}\ …
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3
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0
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249
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Existence of solutions to a reaction-diffusion problem.
Consider the problem of finding a bounded classical solution $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ (such that $u$ is continuous and $u_t$, $u_x$ and $u_{xx}$ exist and are continuous on $\mathbb{R}\ …
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1
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Integral representation of the Cauchy problem solution for the heat equation
I suggest looking at the maximum principles contained in http://rspa.royalsocietypublishing.org/content/470/2167/20140079
and the references therein, which considers uniqueness results for solutions t …
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2
answers
212
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Reference Request: Spatially inhomogeneous solutions to parabolic PDE with homogeneous initi...
I am interested in spatially inhomogeneous classical bounded solutions $u:\mathbb{R}^n \times [0,T] \to \mathbb{R}$ to the Cauchy problem for semi-linear parabolic PDE, which have homogeneous initial …
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LINEAR Parabolic equations. Smooth dependence from initial data
If you wish to consider continuous dependence for your problem, you must further specify the type of solutions you are interested in (i.e. solutions that satisfy a particular bound), for otherwise non …
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Elliptic Harnack inequality for 1D Schrodinger operator?
There is a nice presentation of Harnack inequalities for linear elliptic p.d.e in Protter and Weinberger "Maximum Principles in Differential Equations". Moreover, there are additional references to th …
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Heuristics for boundary Harnack inequality
An illustration of the proof of an associated Harnack inequality for elliptic pdi is given in the book "The Maximum Principle" by Pucci and Serrin (see Chapter 7). Hopefully that helps. I can pull up …
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