<abstract xmlns="http://www.w3.org/1999/xhtml">The uniqueness of classical solutions to parabolic semilinear problems together with nonlocal initial conditions with integrals, for the operator <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_2353737XCT.17.013.6110_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mfrac><mo>∂</mo><mrow><mo>∂</mo><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub></mrow></mfrac><mrow><mo>(</mo><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>t</mi></mrow><mo>)</mo></mrow><mfrac><mo>∂</mo><mrow><mo>∂</mo><msub><mrow><mi>x</mi></mrow><mi>j</mi></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mi>c</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>t</mi></mrow><mo>)</mo></mrow><mo>-</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>,</mo></mrow></math><tex-math>\sum\limits_{i,j = 1}^n {{\partial \over {\partial {x_i}}}\left( {{a_{ij}}\left( {x,t} \right){\partial \over {\partial {x_j}}}} \right)} + c\left( {x,t} \right) - {\partial \over {\partial t}},</tex-math></alternatives></inline-formula>, <italic>x</italic>=(<italic>x</italic>1,...,<italic>x</italic>n), in the cylindrical domain <italic>D</italic>:<italic>D</italic>0 ×(<italic>t</italic>0, <italic>t</italic>0+T) ⊂ ℜn+1, where <italic>t</italic>0 ∈ ℜ, 0 &lt; <italic>T</italic> &lt; ∞, are studied. The result requires that the nonlocal conditions with integrals be introduced.</abstract>

The uniqueness of classical solutions to parabolic semilinear problems together with nonlocal initial conditions with integrals, for the operator ∑i,j=1n∂∂xi(aij(x,t)∂∂xj)+c(x,t)-∂∂t,\sum\limits_{i,j = 1}^n {{\partial \over {\partial {x_i}}}\left( {{a_{ij}}\left( {x,t} \right){\partial \over {\partial {x_j}}}} \right)} + c\left( {x,t} \right) - {\partial \over {\partial t}},, x=(x1,...,xn), in the cylindrical domain D:D0 ×(t0, t0+T) ⊂ ℜn+1, where t0 ∈ ℜ, 0 < T < ∞, are studied. The result requires that the nonlocal conditions with integrals be introduced.

On the uniqueness of solutions to parabolic semilinear problems under nonlocal conditions with integrals

Faculty of Physics, Mathematics and Computer Science

Faculty of Electrical and Computer Engineering

Technical Transactions

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 Public License.

The uniqueness of classical solutions to parabolic semilinear problems together with nonlocal initial conditions with integrals, for the operator...

On the uniqueness of solutions to parabolic semilinear problems under nonlocal conditions with integrals

Published Online: May 23, 2020

Page range: 155 - 163

DOI: https://doi.org/10.4467/2353737XCT.17.013.6110

Keywords
parabolic problems, semilinear equation, nonlocal initial condition with integral, cylindrical domain, uniqueness of solutions

© 2017 Ludwik Byszewski et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 Public License.

On the uniqueness of solutions to parabolic semilinear problems under nonlocal conditions with integrals

Published Online: May 23, 2020

Page range: 155 - 163

DOI: https://doi.org/10.4467/2353737XCT.17.013.6110

Keywordsparabolic problems, semilinear equation, nonlocal initial condition with integral, cylindrical domain, uniqueness of solutions

© 2017 Ludwik Byszewski et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 Public License.

Keywords
parabolic problems, semilinear equation, nonlocal initial condition with integral, cylindrical domain, uniqueness of solutions