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- W2129972893 abstract "The authors prove that the nonlinear parabolic partial differential equation<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=block alttext=[ frac{{partial u}} {{partial t}} = sumlimits_{i,j = 1}^n {frac{{partial ^2 }} {{partial x_i partial x_j }}varphi _{ij} left( u right) - fleft( u right)} ]><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mo>∂</mml:mo><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>∂</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:mfrac><mml:msub><mml:mi>φ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>with homogeneous Dirichlet boundary conditions and a nonnegative initial condition has a nonnegative generalized solution<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$u$><mml:mi>u</mml:mi></mml:math>. They also give necessary and sufficient conditions on the constitutive functions<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$varphi _{ij} $><mml:msub><mml:mi>φ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math>and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$f$><mml:mi>f</mml:mi></mml:math>which ensure the existence of a time<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$t_0 > 0$><mml:msub><mml:mi>t</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:math>for which<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$u$><mml:mi>u</mml:mi></mml:math>vanishes for all<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$t geqslant t_0 $><mml:mi>t</mml:mi><mml:mo>≥</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math>." @default.
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- W2129972893 title "Anisotropic nonlinear diffusion with absorption: existence and extinction" @default.
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